AUTOMORPHIC FORMS ON GL2

This is an introductory course to modular forms, automorphic forms and automorphicrepresentations.

(1) Modular forms(2) Representations of GL2(R)(3) Automorphic forms on GL2(R)(4) Adeles and ideles(5) Representations of GL2(Qp)(6) Automorphic representations of GL2(A)

This is a set of notes for my class ”Automorphic forms on GL(2)” in the University ofChicago, Spring 2011. There is obviously no originality in the content and presentation ofthis very classical materials.

1. Modular forms

As usual in representation theory, the letter G is overused . In each chapter, G will denotea different group. In this chapter G = SL2(R), K = SO2(R), H = G/H is the upper half-plane, D is the open unit disc. Γ will denote a discrete subgroup of SL2(R), Γ its image inPGL2(R). In particular, Γ(1) = SL2(Z) and Γ(1) is its image in PGL2(R).

1.1. Geometry of the upper half-plane. The points of projective line are one-dimensionalsubspaces of a given two-dimensional vector space. The group GL2 of linear transformationsof that two-dimensional vector space thus acts on the corresponding projective line. Theaction of a 2× 2-matrix is given the formula of homographic transformation

(1.1.1)

[a bc d

]z =

az + b

cz + d

if z denotes the standard coordinate of P1. This formula is valid for any coefficients fields.In particular, GL2(R) acts on P1(R) and GL2(C) acts compatibly on P1(C). It follows thatGL2(R) acts on the complement of the real projective line inside the complex projective line

P1(C)−P1(R) = H ∪H−

where H (resp. H−) is the half-plane of complex number with positive (resp. negative)imaginary part. Let GL+

2 (R) denote the subgroup of GL2(R) of matrices with positivedeterminant; it is also the neutral component of GL2(R) with respect to the real topology.Since GL+

2 (R) is connected, its action on P1(C) − P1(R) preserves H and H−. Of course,the above assertion is a consequence of the formula

(1.1.2) =(az + b

cz + d

)=

ad− bc|cz + d|2

=(z).

1

which derives from a rather straightforward calculation

az + b

cz + d=

(az + b)(cz + d)

|cz + d|2

=bd+ aczz + bc(z + z) + (ad− bc)z

|cz + d|2.This equation becomes even simpler when we restrict to the subgroup SL2(R) of real

coefficients matrix with determinant one

(1.1.3) =(az + b

cz + d

)==(z)

|cz + d|2.

From now on in this chapter, we will set G = SL2(R).

Lemma 1.1.1. The group G acts simply transitively on the upper half-plane H. The isotropygroup of the point i ∈ H is the subgroup K = SO2(R) of rotations :

(1.1.4) kθ =

[cos θ sin θ− sin θ cos θ

]Proof. The equation

ai+ b

ci+ d= i

implies that a = d, b = −c in which case the determinant condition ad − bc = 1 becomesa2 + b2 = 1. Thus the matrix is of the form (1.1.4).

Let z = x+ iy with x ∈ R and y ∈ R+. It is enough to prove that there exists a, b, c, d ∈ Rwith ad− bc = 1 such that

ai+ b

ci+ d= z.

We set c = 0. We check immediately that the system of equations ad = 1, a = yd, b = xdhas real solutions with d = y−1/2, a = y1/2 and b = xy−1/2. We observe that this calculationshows in fact G = BK where B is the subgroup of G consisting of upper triangular matrices.This is a particular instance of the Iwasawa decomposition.

Lemma 1.1.2. The metric

(1.1.5) ds2 =dx2 + dy2

y2

on H, as well as the density µ = dxdy/y2 is invariant under the action of G.

Proof. With the notations γ =

[a bc d

]and z′ = γz, we have

(1.1.6) dz′ =(ad− bc)(cz + d)2

dz.

This calculation has the following concrete meaning. The smooth application g : H →H maps z 7→ z′. It induces linear application on tangent spaces TzH → Tz′H and itsdual linear application T∗z′H → T∗zH. The cotangent space T∗z′H (resp. T∗zH) is a one-dimensional C-vector space generated by dz′ (resp. dz). The linear application sends dz on((ad− bc)/(cz + d)2)dz.

2

The element dz induces the canonical quadratic form dx2 + dy2 on TzH viewed as 2-dimensional real vector space. Similarly, we have the quadratic form dx′2 + dy′2 on Tz′H.The equation (1.1.6) implies that

dx′2

+ dy′2

=(cz + d)2

|cz + d|4(dx2 + dy2).

It follows that the metric ds2 = (dx2 + dy2)/y2 is invariant under G, according to (1.1.2).The same argument applies to the density µ = dxdy/y2.

Lemma 1.1.3. The Cayley transform

(1.1.7) z 7→ cz =

[1 −i1 i

]z =

z − iz + i

.

maps isomorphically H onto the unit disk D = z ∈ C | |z| < 1. The inverse transformationis

(1.1.8) w 7→ c−1w =1

2

[1 1i −i

]w =

i(1 + w)

1− w.

The metric ds2 = (dx2 + dy2)/y2 on H transports on the metric

(1.1.9) dDs2 =

4(du2 + dv2)

(1− |w|2)2

where w = u+ iv. We also have

(1.1.10) dxdy/y2 =4dudv

(1− |w|2)2.

Proof. See [5, Lemma 1.1.2] Since c and c−1 are inverse functions of each other, it is enoughto check that c(H) ⊂ D and c−1(D) ⊂ H. For every z ∈ H, we have |z − i| < |z + i| so that|c(z)| < 1. It follows that c(H) ⊂ D. For every w ∈ D, the straightforward calculation

(1.1.11)i(1 + w)

1− w=−2=(w) + i(1− |w|2)

|1− w|2

shows

(1.1.12) y =1− |w|2

|1− w|2> 0.

if z = c−1w and y = =(z). It follows that c−1(D) ⊂ H.By using the chain rule we have

dz =2idw

(1− w)2.

If we write w = u+ iv in cartesian coordinates, then we have

dx2 + dy2 =4(du2 + dv2)

|1− w|4.

It follows thatdx2 + dy2

y2=

4(du2 + dv2)

(1− |w|2)2.

The same calculation proves the expression of the measure on the disc (1.1.10). 3

Lemma 1.1.4. Any two points of H are joined by a unique geodesic which is a part of acircle orthogonal to the real axis or a line orthogonal to the real axis.

Proof. See [5, Lemma 1.4.1]. Instead of H we consider the unit disc. We assume that thefirst point is 0 and the second point is a positive real number a < 1. Let φ : [0, 1]→ D withφ(t) = (x(t), y(t)) denote a parametrized joining 0 = (x(0), y(0)) and a = (x(1), y(1)). Itslength is ∫ 1

0

2(1− |φ(t)|2)−1√

(dx(t)/dt)2 + (dy(t)/dt)2dt

that is at least ∫ 1

0

2(1− x(t)2)−1|dx(t)/dt|dt =

∫ a

0

2dt

(1− t2)

The shortest curve joining 0 and a is thus a part of a radius in the unit disc.For every two points x0, x1, there is g ∈ SL2(R) that maps H on D, cg(x0) = 0 and

cg(x1) = a where a is a positive real number satisfying a < 1. Here c : H → D is theCayley transform. The geodesic joining x0 with x1 is a part of the preimage of the radiusfrom 0 to a. That preimage is necessarily part of a circle or a strait line. Moreover as thetransformation cg is conformal, that circle or line must be orthogonal with the real line asthe radius [0, a] is orthogonal to the unit circle.

Exercice 1.1.5. [2, Ex. 1.2.5] Let SL(2,C) acts of bP 1(C) by the homographic transforma-tion (1.1.1). Prove that the subgroup that map the unit disc D onto itself is

(1.1.13) SU(1, 1) =

[a bb a

]| |a|2 − |b|2 = 1

.

Prove that the subgroup SU(1, 1) is conjugate to SL(2,R) in SL(2,C). Prove that the subgroupof SU(1, 1) that fixes 0 ∈ D is the rotation group[

eiθ 00 e−iθ

].

1.2. Fuschian groups. We will be mainly interested on the quotient of H by a discretesubgroup of G. The most important examples of discrete subgroups are the modular groupSL2(Z) and its subgroup of finite indices. We will call Fuchsian group a discrete subgroupof SL2(R).

Proposition 1.2.1. A Fuchsian group Γ acts properly on the upper half-plane H.

Proof. Recall that the action Γ on H is proper means that the map Γ×H→ H×H definedby (γ, x) 7→ (x, γx) is proper i.e. the preimage of a compact is compact. We need to provethat for every compact subsets U, V ⊂ H, the set γ ∈ Γ|γU ∩ V 6= ∅ is a finite. Becausethe group G = SL2(R) acts on H with compact stabilizer, the subset g ∈ G|γU ∩ V 6= ∅is compact. Its intersection with the discrete subgroup Γ is finite.

Corollary 1.2.2. For every Fuchsian group Γ, the quotient Γ\H is a Haussdorf topologicalspace.

4

An element g ∈ SL2(R) is called elliptic if it has a fixed point in H. It follows from theabove that g is elliptic if and only if it is conjugate to an element of SO(2,R)[

cos θ sin θ− sin θ cos θ

]Any element g ∈ SL2(C) has at least on fixed point in P1(C). If g ∈ SL2(R) is not elliptic,it must have fixed points on P1(R) = R ∪ ∞. We call g parabolic if it has a unique fixedpoint, and hyperbolic if it has two fixed points. A parabolic element is conjugate to a matrixof the form

(1.2.1)

[ε x0 ε

]with ε ∈ ±1. A hyperbolic element is conjugate to a diagonal matrix

(1.2.2)

[a 00 a−1

]with a ∈ R×.

Let Γ be a Fuschian group. For every x ∈ H, the stabilizer Γx of x in Γ is a finite groupbecause it is the intersection of a compact group with a discrete group. In fact, since SO2(R)is isomorphic to the circle, for every x ∈ H, Γx is a finite cyclic group. If Γx is nontrivial,we call x an elliptic point of Γ. This also implies taht elliptic points are isolated.

Lemma 1.2.3. There is a canonical complex structure on Γ\H so that the quotient mapH→ Γ\H is complex analytic.

Proof. If [z] is the Γ-orbit of z ∈ H such that Γz = Γ∩Z. Then there exists a neighborhoodU of z consisting of points with the same property. This neighborhood is homeomorphic withits image in Γ\H. Its image in Γ\H is thus equipped with an analytic structure inheritedfrom U .

If z is the Γ-orbit of z ∈ H such that Γx = Γ ∩ Z such that Γz is a finite group largerthan Γ ∩ Z. Γz is then a finite cyclic group µk. By a homographic transformation we canchange the model from H to D and maps z to 0. The quotient of a small disc around 0 bythe action of µk can be given a complex structure with uniformizing parameter wd where wis the standard uniformizing parameter of D around 0. This defines a complex structure ona neighborhood of [z] in Γ\H.

Definition 1.2.4. A point x ∈ P1(R) is called a cusp for Γ if it is the fixed point of anontrivial parabolic element.

In that case Γx is isomorphic to the product of Z with an infinite cyclic group, the firstfactor Z = µ2 being the center of G. Let PΓ denote the set of cusps of Γ. We set

H∗ = H ∪ PΓ.

We consider the topology on H∗ by adding to the real topology on H a family of neighbor-hoods of each cusp x ∈ PΓ. If x =∞, we take the family

(1.2.3) U∗l = Ul ∪ ∞ with Ul = z ∈ H | =(z) > lThe family of neighborhoods of other cusps are constructed from the Ul by conjugation.

Lemma 1.2.5. Γ\H∗ is a Haussdorff space.5

Proof. See [5, Lemma 1.7.7]. As we already know that Γ\H is Haussdorff, it remains toprove that a cusp and a point of H are separated and two cusps are separated that can bechecked directly upon the definition.

Lemma 1.2.6. There is a complex analytic structure on Γ\H∗ that extends the complexanalytic structure on Γ\H.

Proof. We can restrict ourselves to the case that ∞ is a cusp and to define an analyticstructure around the image of ∞ in Γ\H∗. The stabilizer of ∞ in G is

(1.2.4) G∞ = [a b0 a−1

]|a ∈ R×, b ∈ R

By definition, ∞ is a cusp of Γ if Z(Γ ∩G∞) is a subgroup of the form

(1.2.5) [ε mn0 ε

]|ε ∈ ±1,m ∈ Z

for some fixed integer n. The map z 7→ e2iπz/n defines a homeomorphism from (G∞ ∩Γ)\U∗lwhere U∗l is a standard neighborhood (1.2.3) of ∞ ∈ H∗ on a disc centered at 0. Thisprovides (G∞ ∩ Γ)\H∗ with a complex analytic structure.

Definition 1.2.7. A discrete subgroup Γ of SL2(R) is called a Fuchsian group of first kindif XΓ = Γ\H∗ is compact.

Proposition 1.2.8. If XΓ is compact, then the numbers of elliptic points and cusps of Γ inΓ\H are finite.

Theorem 1.2.9 (Siegel). A discrete subgroup of SL2(R) is a Fuchsian group of first kind ifand only if Γ\H has finite area.

Proof. We refer to [5, Theorem 1.9.1] for the proof of this theorem. We will only be interestedin the case of arithmetic groups in which the conclusion of the theorem can be establisheddirectly by other means.

A connected domain F of H is called a fundamental domain of Γ if F satisfies the followingconditions

(i) H =⋃γ∈Γ γF ;

(ii) if U is the set of interior points of F then F = U ;(iii) γU ∩ U = ∅ for all γ ∈ Γ not belonging to the center Z of G.

Lemma 1.2.10. Every Fuchsian group has a fundamental domain.

Proof. An element γ ∈ Γ−Z only has finitely many fixed points. Since Γ is countable, thereexists z0 ∈ H which is not fixed by any element γ ∈ Γ− Z. For every γ ∈ Γ, we put

Fγ = z ∈ H | d(z, z0) ≤ d(z, γz0)Uγ = z ∈ H | d(z, z0) < d(z, γz0)Cγ = z ∈ H | d(z, z0) = d(z, γz0)

Here, d indicates the hyperbolic distance on H defined by the metric ds2 = (dx2 + dy2)/y2.The intersection

F =⋂

γ∈Γ−Z

Fγ

6

is a fundamental domain of Γ.

We will now review the classification of 2 × 2 real matrices up to conjugation. A matrixis said to be :

(1) hyperbolic if its has distinct real eigenvalues;(2) elliptic if it has distinct complex conjugate eigenvalues;(3) parabolic if it is not central and and has an eigenvalue of multiplicity two;(4) central otherwise.

Lemma 1.2.11. Let γ ∈ SL2(Q) act on P1(C) by homographic transformation. Let z ∈ Cso that γz = z. If γ is hyperbolic (resp. elliptic) then z is either rational or generates a real(resp. imaginary) quadratic extension of Q. If γ is parabolic then z ∈ Q.

Among the Fuschian groups, we are particularly interested in the modular group SL2(Z)and its subgroups of congruence

Γ(N) = γ ∈ SL(2,Z) | γ ≡ 1 mod N(1.2.6)

Γ1(N) = γ ∈ SL(2,Z) | γ ≡[1 ∗0 1

]mod N(1.2.7)

Γ0(N) = γ ∈ SL(2,Z) | γ ≡[∗ ∗0 ∗

]mod N(1.2.8)

In particular, we will use the convenient notation Γ(1) = SL2(Z) for the full modular group.We consider the case of the full modular group. Let F denote the domain defined by the

conditions |<(z) ≤ 1/2| and |z| ≥ 1. Consider the two matrices of Γ(1)

(1.2.9) T =

[1 10 1

]and S =

[0 −11 0

]They act on H by the following rules Tz = z + 1 and Sz = −1/z.

Lemma 1.2.12. Let Γ′ denote the subgroup of Γ(1) generated by the transformations S andT as above.

(1) For every z ∈ H, there exists γ ∈ Γ′ such that γz ∈ F .(2) If z, z′ ∈ F and γ ∈ Γ(1) non trivial such that γz = z′ then z, z′ both lies in the

boundary of F .(3) F is a fundamental domain for Γ(1), and Γ(1) is generated by the matrices S and T .

Proof. For every z ∈ H, the lattices generated 1 and z have only finitely many memberscz + d such that |cz + d| ≤ 1. It implies that there are only finitely many z′ = γz conjugateto z such that =(z′) ≥ =(z). We can assume that =(z) is maximal among all the conjugatesγz with γ ∈ Γ′. With the help of the translation T , we can assume that z belongs the thevertical strip <(z) ≤ 1/2. We only need to prove that under these assumptions, we have|z| ≥ 1. If |z| < 1, we would have =(−1/z) > =(z) that would contradict the maximality of=(z). It follows that z ∈ F .

Let z, z′ ∈ F and

γ =

[a bc d

]∈ Γ(1) such that z′ = γz.

7

We can assume that =(z) ≤ =(z′). This implies that |cz + d| ≤ 1. By a careful inspection,this implies in particular that z must lie on the boundary of F . More inspection shows thatz′ lie also on the boundary of F . See [6, p.130].

Let z ∈ U be an element of the interior of F . For every γ ∈ Γ, there exists γ′ ∈ Γ′ suchthat γ′γz ∈ F . The assumption that z lie in the interior of F implies γ′γ = 1, thus γ ∈ Γ′.We have also checked all the conditions that makes F a fundamental domain of Γ(1).

Proposition 1.2.13. For Γ(1) = SL(2,Z), the set of cusps is Q ∪ ∞. They are allconjugate under the action of Γ(1).

The quotient XΓ(1) = Γ(1)\H∗ is isomorphic to P1(C) as complex analytic space. Up toequivalence, Γ(1) has one elliptic point of order 2 that is i ∈ H with b2 = −1 and one ellipticpoint of order 3 that is j ∈ H with j3 = 1.

Proof. An element x ∈ R which fixed by a parabolic matrix γ ∈ Γ(1) must be a rational.It follows from the shape of the fundamental domain F of Γ(1) that Γ(1)\H∗ is a compactRiemann surface that is homeomorphic to the sphere. It is isomorphic to P1(C).

Corollary 1.2.14. Congruence subgroups are Fuchsian groups of first kind who set of cuspsis Q ∪ ∞. There are only finitely many cusps up to the action of Γ.

Proof. Since Γ is a subgroup of Γ(1) with finite index, they have the same set of cuspsQ ∪ ∞. Since Γ(1) acts transitively on this set, the number of Γ-orbits in this set is atmost equal to the index of Γ in Γ(1).

Since Γ(1)\H∗ is compact, and Γ is a subgroup of Γ(1) of finite index, the quotient Γ\H∗is also compact. Let xn be a sequence of points of Γ\H∗. We need to prove that there existsa convergent subsequence. Let zn be a sequence in H∗ so that xn = Γzn is the image of zn inΓ\H. Let xi denote the image of zn in Γ(1)\H∗. Since Γ(1)\H∗ is compact, we can assumethat xn converges the x ∈ Γ(1)\H∗. Let z ∈ H∗ be a preimage of x. There exists γn ∈ Γ(1)so that γnzn converges to z. Because Γ/Γ(1) is finite, after extracting a subsequence, we canassume that there exist γ ∈ Γ(1) so that γn ∈ γΓ(1) for all n. It follows that xn convergesto γ−1x where x is the image of z in Γ\H∗. This proves that Γ is a Fuschian group of firstkind.

Proposition 1.2.15. Let Γ be a subgroup of Γ(1) of finite index µ. Let m2,m3 be the numberof Γ-equivalence classes of elliptic points of orders 2 and 3 respectively. Let m∞ denote thenumber of Γ-equivalence classes of cusps. Then the genus of Γ\H∗ is

g = 1 +µ

12− m2

4− m3

3− m∞

2.

Proof. See [7, 1.40]. The mapΓ\H∗ → Γ(1)\H∗

is a finite proper map of degree µ = [Γ(1) : Γ].This is an application of Hurwitz’ formula. As the morphism Γ\H∗ → Γ(1)\H∗ is of

degree µ and Γ(1)\H∗ is of genus 0, we have

2g − 2 = −2µ+∑P

(eP − 1)

with P in the set of ramified points, eP being the index of ramification. Summing eP overthe ramified points P over j we get 2(µ −m3)/3. The same sum over i is (µ −m2)/2 and

8

over ∞ is µ − m∞. By summing altogether, we get the desired formula for the genus ofΓ\H∗.

Corollary 1.2.16. If Γ do not have elliptic points then the genus of Γ\H is

g = 1 +µ

12− m∞

2.

Exercice 1.2.17. [2, p.24]

(1) Prove that a fundamental domain for Γ(2) consists of x + iy such that −1/2 < x <3/2, |z + 1/2| > 1/2, |z − 1/2| > 1/2 and |z − 3/2| > 1/2.

(2) Prove that Γ(2) is generated by the matrices[1 20 1

]and

[1 02 1

](3) Prove that Γ(2) has three inequivalent cusps and Γ(2)\H∗ is isomoprhic to P1(C)(4) Prove that if φ is an entire function such that there exists two distinct complex num-

bers a, b that don’t belong to the image of φ then φ is a constant function (Picard’stheorem).

1.3. Modular forms. Let k be an even nonnegative integer. A modular form of weight kfor Γ = SL(2,Z) is a holomorphic function on H which satisfies the identity

(1.3.1) f

(az + b

cz + d

)= (cz + d)kf(z)

for all z ∈ H and [a bc d

]∈ SL(2,Z)

and which is holomorphic at the cusp ∞. The last condition requires some discussion. Wehave define the analytic structure of Γ\H∗ near ∞ by choosing as the local coordinate thefunction q = e2πiz. The equation 1.3.1 implies in particular f(z + 1) = f(z), and thus f hasa Fourier expansion

(1.3.2) f(z) =∑n∈Z

ane2iπnz =

∑n∈Z

anqn.

The function f is holomorphic at the cusp ∞ if in the above expansion an = 0 for n < 0. Iffurthermore a0 = 0, we say that f is cuspidal at ∞.

If Γ is a Fuschian group of first kind, we can also modular forms of weight k for Γ similarly.The holomorphic function f on H is required to satisfy the same equation (1.3.1) and to beholomorphic at the cusps of Γ. If x is a cusp, the stabilizer of x in PGL2(Z) is the infinitecyclic group generated by a parabolic element. After conjugation, we can assume that thecusp is the point ∞ and its stabilizer in Γ is generated by the matrix

(1.3.3)

[1 m0 1

]for some positive real number m ∈ R. The holomorphicity at this cusp is equivalent to thatf admits a Fourier expansion f =n∈Z anq

n with an = 0 for n < 0 with respect to the variableq = e2iπz/m.

9

Lemma 1.3.1. Suppose that ∞ is a cusp of a Fuschian group Γ of first kind with Γ∞generated by the matrix (1.3.3). Let f be a modular form of weight k and let

∑anq

n withq = e2iπz/m denote its Taylor expansion near this cusp. Then the series

∑anq

n convergesabsolutely and uniformly on every compact in H.

Proof. The function z 7→ q = e2iπz/m defines an isomorphism between Γ∞\H and the punc-tured disc D − 0. By assumption modular form f defines a holomorphic function onD− 0 that extends holomorphically to D. This implies that the Taylor series

∑∞n=0 anq

n

converges absolutely uniformly on every compact contained in D.

Definition 1.3.2. Let Γ be a Fuschian group of first kind. We denote Mk(Γ) the space ofmodular forms of weight k for Γ. We denote by Sk(Γ) the space of cusp forms of weight kfor Γ. We also denote Ak(Γ) the space of meromorphic functions f of H satisfying (1.3.1)that are meromorphic at the cusps.

Proposition 1.3.3. If k = 0, A0(Γ) is the field FΓ of meromorphic functions on XΓ. Wehave M0(Γ) = C and S0(Γ) = 0.

We will now determine the dimension of Mk(Γ) and Sk(Γ) for even integers k. We willrefer to [7, 2.6] and [5] for the case of odd integers. The case k = 1 does not seem to betreated so far.

Proposition 1.3.4. There is a canonical isomorphism between the space A2k(Γ) of mero-morphic automorphic forms of weight 2k and the space Ω⊗kXΓ

⊗OXΓFΓ of meromorphic k-fold

differential form on XΓ. In particular, A2k(Γ) is a one-dimensional FΓ-vector space.

Proof. For every f ∈ Ak(Γ), the k-fold differential form f(z)(dz)⊗k is Γ-invariant. It descendsto a meromorphic k-fold differential form ωf on Γ\H. The condition of meromorphicity of fat the cusps impies that ωf is a meromorphic form on XΓ. The application f 7→ ωf inducesan isomorphism A2k(Γ)→ Ω⊗kXΓ

⊗OXΓFΓ.

In order to calculate the dimension of Mk(Γ), we will express the condition of holomorphic-ity of f in terms of the zero divisor of ωf on XΓ and then apply the theorem of Riemann-Roch.This calculation will be done separately in three cases : general points, elliptic points andcusps :

• Let z ∈ H be a non-elliptic point with image x ∈ XΓ. The function f is holomorphicat z0 if and only if ωf is holomorphic at x.• Let z ∈ H be an elliptic point of index e and let denote x ∈ XΓ its image. Let t

be a local parameter at z ∈ H and u a local parameter at x ∈ XΓ. We have te ∼ uwhere the equivalence means equal up to an invertible function on a neighborhoodof z. By derivation, we have du ∼ ze−1dz. Raising to the power k, we have (dz)⊗k ∼z−k(e−1)(du)⊗k. Let denote νx(f) the valuation of f with respect to the parameter ui.e. νx(u) = 1 and νx(t) = 1/e. Let us calculate the order of vanishing of the k -folddifferential form

ωf = f(dz)⊗k ∼ fz−k(e−1)(du)⊗k

at x. We haveordx(ωf ) = νx(f)− k(1− 1/e).

The function f is holomorphic at z if and only if νx(f) ≥ 0 which is equivalent to

ordx(ωf ) + k(1− 1/e) ≥ 0.10

Since ordx(ωf ) is an integer, the above inequality is equivalent to

ordx(ωf ) + [k(1− 1/e)] ≥ 0

where as usual [r] denotes the largest integer that is not greater than a given realnumber r. In the case of weight two form i.e. k = 1, the above condition meanssimply that ordx(ωf ) ≥ 0. In the general case the integer e cannot be ignored.• Let us consider a cusp of Γ that we can assume to be ∞ without loss of generality.

Let us denote x its image in Γ\XΓ. The development of f at the cusp has the form

f(z) =∑n∈Z

anqn

where q = 2iπmz for some positive integer m. Let r be the least integer such thatar 6= 0. We note νx(f) = r. Let us denote ωf = fdz⊗k. Since dz ∼ dq/q we haveωf ∼ fq−kdq⊗k. By construction, q is a local parameter of XΓ at x. It follows that

νx(ωf ) = νx(f)− k.

Thus f is holomorphic at the cusp ∞ i.e. νx(f) ≥ 0 if and only if

νx(ωf ) + k ≥ 0

and f vanishes at the cusp ∞ i.e. νx(f) ≥ 1 if and only if

νx(ωf ) + k ≥ 1.

In wight two case k = 1, f is holomorphic at ∞ if ωf is a logarithmic one form andf is a cusp form if and only if ωf is a holomorphic one form.

Let denote x its image in Γ\H and let choose a local parameter u of x ∈ Γ\H∗.

Proposition 1.3.5. Let Γ be a Fuschian group of first kind. The space M2(Γ) is canonicallyisomorphic with the space H0(XΓ,ΩXΓ

(cusp)) of one form with logarithmic singularities atthe cusps. The space S2(Γ) is canonically isomorphic with the space of holomorphic one formof XΓ

S2(Γ) = H0(XΓ,ΩXΓ).

In particular dimS2(Γ) = g (calculated in 1.2.15) and dimM2(Γ) = g+m− 1 where g is thegenus of XΓ and m is the number of inequivalent cusps.

Proposition 1.3.6. Let Γ be a Fuschian group of first kind and let 2k be an even integergreater or equal to 4. We have

dimM2k(Γ) = dimS2k(Γ) +m.

and

dimS2k(Γ) = (2k − 1)(g − 1) +s∑i=1

[k(1− 1/ei)] + (k − 1)

where x1, . . . , xs denote the elliptic points, e1, . . . , es their elliptic index and m is the numberof inequivalent cusps. In absence of elliptic points, we have

dimS2k(Γ) = (2k − 1)(g − 1) + (k − 1)m.

In the case Γ = Γ(1), we have g = 0, m = 1 and two elliptic points with indexes 2, 3.11

Corollary 1.3.7. We have

dimS2k(Γ(1)) =

0 if k = 1

[k/6]− 1 if k > 1 and k ≡ 1 mod 6

[k/6] otherwise

and

dimM2k(Γ(1)) =

0 if k = 2

dimS2k(Γ(1)) + 1 otherwise

In particular dimM4(Γ(1)) = dimM6(Γ(1)) = 1. We can construct an explicit generatorfor these spaces by Eisenstein series. Let 2k be an even integer with 2k ≥ 4. Define

(1.3.4) E2k(z) =1

2

∑(m,n)∈Z2−(0,0)

(mz + n)−2k

This series is absolutely uniformly convergent on compact domain and defines a holomorphicfunction on H. This function is a modular form of weight 2k for the full modular group Γ =SL(2,Z). The automorphy (1.3.1) dervies from the action of SL(2,Z) on the set Z2 − (0, 0).We will prove in 1.4.1 that Eisenstein are holomorphic at the cusp. In particular, it will beproved that the free coefficient in the Fourier expansion of E2k is ζ(2k). We will choose anormalization so that the free coefficient be one

G2k(z) = ζ(2k)−1E2k.

The space M4(Γ(1)) is generated by G4, M6(Γ(1)) is generated by G6, M8(Γ(1)) is gener-ated by G2

4, M10(Γ(1)) is generated by G4G6. In weight 12 there is the first cusp form

∆ = (G34 −G2

6)/1728.

Proposition 1.3.8. The rational function j : G34/∆ defines an isomorphism from XΓ(1) onto

P1C

Proof. By the discussion that precedes 1.3.5, there is a line bundle L on XΓ(1) such thatM12(Γ(1)) = H0(XΓ(1),L). We know dim H0(XΓ(1),L) = 2. Since XΓ(1) = P1, L = OP1(1).It follows that G3

4 and ∆ as global section of L vanish exactly at one point. Morever as theyare not proportional, their quotient define a morphism

j : G34/∆ : XΓ(1) → P1

C

that is an isomorphism.

1.4. Fourier coefficients of modular forms. We have an explicit formula for Fouriercoeffients of Eiseinstein series

Proposition 1.4.1. The Fourier expansion of Ek has the form

(1.4.1) Ek(z) = ζ(z) +(2πi)k

(k − 1)!

∞∑n=1

σk−1(n)qn.

where σk−1(n) =∑

d|n dk−1.

12

Proof. See [2, p.28] The terms with m = 0 in LHS sum up to the term ζ(z) in RHS. We have

1

2

∑n∈Z−0

n−k =∑n∈N

n−k = ζ(k).

In order to deal with the other terms, we will need the following lemma.

Lemma 1.4.2. Let k be an integer greater or equal to two. We have the formula

(1.4.2)∑n∈Z

(n− z)−k =(2πi)k

(k − 1)!

∑n∈N

nk−1e2πinz

for all z ∈ H.

Proof. See [2, p.12] for more details. For a fixed z, the function f(x) = (x−z)−k is a complexanalytic function with a pole at x = z. On the real line, it has no pole if =(z) > 0 and it isL1 if k ≥ 2. Its Fourier transform is given by the formula

f(y) =

∫ ∞−∞

(x− z)−ke2iπxydx.

We can evaluate this integral by applying the residue formula to the 1-form (x−z)−ke2iπxydx.We get

f(y) =

2πi resx=z((x− z)−ke2πixydx) if y > 0

0 if y ≤ 0.

The calculation of the residue gives

f(y) =

(2πi)k

(k−1)!yk−1e2πiyz if y > 0

0 if y ≤ 0.

We apply now the Poisson summation formula reviewed in B.2.2.

In (1.3.4), the terms with a fix m > 0 and and thoese with its opposite are equal. Bytaking the factor 1/2 into account, we only need to consider the terms with m > 0. Applythe above lemma to mz, we will get

(1.4.3)∑n∈Z

(mz + n)−k =(2πi)k

(k − 1)!

∑n∈N

nk−1e2πimnz.

If we sum the above formula over the positive integers m, we will get (1.4.1).

Corollary 1.4.3. Let Ek(z) =∑∞

n=0 anqn be the Fourier expansion of the Eisenstein series

at ∞. There exists positive constants A,B > 0 such that Ank−1 ≤ an ≤ Bnk−1 for everyn ∈ N.

Proof. By the formula (1.4.1), it is enough to seek such an estimation for σk−1(n). In one sidewe have the obvious inequality nk−1 ≤ σk−1(n). On the other hand, we have the inequality

σk−1(n)

nk−1=∑d|n

1

dk−1≤ ζ(k − 1)

valid for k > 2. 13

Proposition 1.4.4. If f =∑∞

n=1 anqn is a cusp form of weight k, its Fourier coefficients

satisfy the inequality an ≤ Cnk/2 for some constant C independent of n.

Proof. The equations (1.1.2) and (1.3.1) imply that the continuous function |f(z)yk/2| onH is Γ-invariant. It so defines a continuous function on the quotient Γ\H. Assume nowΓ = SL2(Z). Since |q| = e−2πy, the vanishing of the constant terms of the Fourier expansionof f in the variable q implies that limy→∞ |f(z)yk/2| = 0. It follows that the function|f(z)yk/2| is bounded by a constant C1. For every natural integer n ∈ N, and for everyy > 0, we have

|an|e−2πny = |∫ 1

0

f(x+ iy)e−πinxdx| ≤ C1y−k/2.

Let us pick y = 1/n and derive the inequality

|an| < Cnk/2

with C = e2πC1.

This bound can be improved according to the Ramanujan-Peterson conjecture.

Theorem 1.4.5. Let f =∑∞

n=1 anqn is a cusp form of weight k of level N . Then for

(n,N) = 1, we have an = O(nk−1

2 ).

This conjecture was proved by Eichler, Shimura and Igusa in the case k = 2. The proofin the k > 2 is due to Deligne. It is based on the Eichler-Shimura relation and the Weilconjecture.

Among cusp forms, the eigenvectors with respect to the Hecke operators that we willlater introduce, have Fourier coefficients with arithmetic significance. In particular, sincethe space S12(Γ(1)) of cusp forms of weight 12 for Γ(1) is one dimensional, its generatoris automatically an eigenvector. The ∆ function of Ramanujan ∆(z) =

∑∞n=1 τ(n)qn is a

normalized cusp form whose Fourier coefficients are integers. Deligne proved the inequality

|τ(p)| ≤ 2p11/2

that is the original conjecture of Ramanujan.

1.5. L-function attached to modular forms. If f ∈ Mk(Γ) is a modular form withFourier expansion f =

∑∞n=1 anq

n. We call the Dirichlet series

(1.5.1) L(s, f) =∞∑n=1

ann−s

The bounds on the Fourier coefficients 1.4.3 and 1.4.4 implies that this Dirichlet seriesconverges on a half-plane. We also consider the complete L-function

(1.5.2) Λ(s, f) = (2π)−sΓ(s)L(s, f).

Hecke’s theory takes a rather simple form in the case of the full modular group Γ(1).

Proposition 1.5.1. Suppose that f is a modular form of weight k for Γ(1). If f is a cuspform, Λ(s, f) extends to an analytic function of s, bounded on vertical strips. If f is nota cusp form, then Λ(s, f) extends to a meromorphic function with simple poles s = 0 ands = k. It satisfies the functional equation

(1.5.3) Λ(s, f) = ikΛ(k − s, f).14

Proof. We will restrict ourselves to the case of a cusp form for the full modular group.Because f is cuspidal, f(iy)→ 0 vary rapidly as y →∞. We use the automorphy equation(1.3.1) for the element

S =

[0 −11 0

]∈ Γ(1)

and derive the equality

(1.5.4) f(iy) = iky−kf(i/y)

It follows that f(iy)→ 0 very rapidly as y → 0 too. It follows that the integral

(1.5.5)

∫ ∞0

f(iy)ysdy

y

is convergent for all s and defines an analytic function of s.The following Mellin integral is absolutely convergent for <(s) > ν + 1∫ ∞

0

f(iy)ysdy

y=

∫ ∞0

∞∑1

ane−2nπyys

dy

y(1.5.6)

=∞∑1

an(2nπ)−s∫ ∞

0

e−yysdy

y(1.5.7)

= (2π)−sΓ(s)∞∑1

ann−s(1.5.8)

= Λ(s, f)(1.5.9)

The exchange of the integration and infinite series is licit because the series∑∞

1 ane−2nπy is

absolutely convergent as well as∑∞

1 ann−s. It follows that the expression 1.5.5 defines an

analytic continuation of Λ(s, f).The functional equation (1.5.3) derives from the substitution of y by 1/y in (1.5.5).

The converse theorem is also easy in the case of Γ(1).

Proposition 1.5.2. Let a1, a2 . . . be a sequence of complex numbers which is O(nν) forsome positive real number ν. Let L(s, f) be defined by the series (1.5.1), convergent for<(s) > ν + 1. Let Λ(s, f) be the function defined by (1.5.2) and suppose it has an analyticcontinuation for the complex plan of s which is bounded in any vertical strips. Assume thatΛ(s, f) satisfies the functional equation (1.5.3) for some positive integer k.

Then f(z) =∑∞

n=1 anqn is a cusp form of weight k for Γ(1).

Proof. The assumption an = O(nν) implies that the series∑∞

n=1 anqn is absolutely for |q| < 1

or over the half-plane z = H if q = e2πiz. The Mellin integral∫ ∞0

f(iy)ysdy

y=

∫ ∞0

∞∑1

ane−2nπyys

dy

y(1.5.10)

=∞∑1

an(2nπ)−s∫ ∞

0

e−yysdy

y(1.5.11)

= Λ(s, f)(1.5.12)15

is absolutely convergent for <(s) > ν+1. Let σ be a positive real number such that σ > ν+1.The inverse Mellin transform will permit us to recover the value of f on the imaginary axisiR from Λ(s, f)

(1.5.13) f(iy) =1

2πi

∫ ∞t=−∞

Λ(σ + it, f)y−σ−itdt.

The convergence of this integral follows from the absolute convergence of the Dirichlet seriesand the Stirling formula for the Gamma function:

(1.5.14) Γ(s) ∼√

2πe−sss−1/2

as |s| → ∞ and <(s) ≥ δ > 0. On the vertical line s = σ + it for fixed σ > 0, we have

|Γ(σ + it)| ∼√

2π|t|σ−1/2e−π|t|/2

as |t| → ∞.With the functional equation (1.5.3), we can transform (1.5.13) as follows

f(iy) = ik1

2πi

∫ ∞t=−∞

Λ(k − σ − it, f)y−σ−itdt(1.5.15)

= iky−k1

2πi

∫ ∞t′=−∞

Λ((k − σ) + it′, f)yk−σ+it′dt′(1.5.16)

after the change of variable t′ = −t. We note that the line <(s) = k − σ is out of theconvergence domain on the Dirichlet series and we would like to move back the line ofintegration to the domain of convergence of the Dirichlet series. In order to apply Cauchytheorem, we need to prove that Λ(x + iy, f) → 0 as y → ±∞ uniformly when x varies ina compact set. For a fixed x > σ, this follows again from the absolute convergence of theDirichlet series and the Stirling formula for the Gamma function.For a fixed x << 0, weobtain the same convergence Λ(x+ iy, f)→ 0 as y → ±∞ by using the functional equation(1.5.3). The uniform convergence follows from an application of the Phragmen-Lindelofprinciple.

Proposition 1.5.3. Let f(s) be a holomorphic function on the upper part of a strip definedby the inequalities

σ1 ≤ <(s) ≤ σ2 and =(s) > c.

Suppose that f(σ+ it) = O(etα) for some real number α > 0. Suppose that for some M ∈ R,

f(σ + it) = O(tM) for σ = σ1 or σ = σ2. Then f(σ + it) = O(tM) uniformly in σ ∈ [σ1, σ2].

Proof. See [5, p.118] and [10, p.124].

We can now apply the Cauchy theorem and move back the integration line to <(s) = σ

(1.5.17) f(iy) = iky−k1

2πi

∫ ∞t=−∞

Λ(σ + it, f)yσ+itdt

and apply again the inverse Mellin transform

(1.5.18) f(iy) = iky−kf(iy−1).

This is the automorphy relation (1.3.1) with respect to the element S of (1.2.9). The auto-morphy relation for T requires no proof since f is defined as a Fourier series. As S and Tgenerates the full modular group Γ(1), f is a cusp form of weight k for Γ(1).

16

The case of a general congruence group Γ ⊂ Γ(1) is considerably more complicated, mainlybecause the matrix S fails to belong to Γ. As we will see later 1.7, it it enough to restrict tocongruence subgroup of the form Γ(N) ⊂ Γ1(N) ⊂ Γ0(N) with

Γ(N) = γ ∈ SL(2,Z) | γ ≡ 1 mod N(1.5.19)

Γ1(N) = γ ∈ SL(2,Z) | γ ≡[1 ∗0 1

]mod N(1.5.20)

Γ0(N) = γ ∈ SL(2,Z) | γ ≡[∗ ∗0 ∗

]mod N(1.5.21)

For every N , there is an exact sequence

1→ Γ1(N)→ Γ0(N)→ (Z/NZ)× → 1.

It follows that the group (Z/NZ)× acts on the space of modular forms Mk(Γ1(N)) as wellas the space of cusp forms Sk(Γ1(N)). For every primitive character χ : (Z/NZ)× → C×, wewill denote

(1.5.22) Mk(N,χ) = f ∈Mk(Γ1, N) | cf = χ(c)f ∀c ∈ (Z/NZ)×.Similar notation Sk(N,χ) for cusp forms will also prevail. We will call these forms modularforms of level N of nebentypus χ.

We will need the following lemma that replaces the role of the element S ∈ Γ(1).

Lemma 1.5.4. The matrix

(1.5.23) SN =

[0 −1N 0

]normalizes Γ0(N). Furthermore, it transforms the space Sk(N,χ) into Sk(N, χ) where χ isthe opposite Dirichlet character of χ.

Theorem 1.5.5 (Hecke). Let f(z) =∑∞

n=0 anqn and g(z) =

∑∞n=0 bnq

n where q = e2πiz

and an, bn are O(nν) for some real number ν. For positive integers k and N , the followingconditions are equivalent

(A) g(z) = (−i)kNk/2z−kf(−1/Nz).

(B) Both ΛN(s, f) = (2π/√N)−sΓ(s)L(s, f) and ΛN(s, g) = (2π/

√N)−sΓ(s)L(s, g) can

be analytically continued to the whole s-plane, satisfy the functional equation

(1.5.24) Λ(s, f) = Λ(k − s, g)

and

ΛN(s, f) +a0

s+

b0

k − sis holomorphic on the s-plane and bounded on any vertical strip.

See [5, 4.3.5] for proof.

Theorem 1.5.6 (Weil). Let N be a positive integer and χ be a Dirichlet character mod-ulo N . Suppose an, bn are sequences of complex numbers such that an, bn = O(nν) forsome real number ν. If D is positive integer number, relatively prime to N , and if µis a primitive Dirichlet character modulo D, we consider the Dirichlet series La(s, µ) =∑∞

n=0 µ(n)ann−s and Lb(s, µ) =

∑∞n=0 µ(n)bnn

−s. Let Λa(s, µ) = (2π)−sΓ(s)La(s, µ) andΛb(s, µ) = (2π)−sΓ(s)Lb(s, µ).

17

Let S be a set of primes including those dividing N . Assuming that the conductor D of µis either 1 or a prime D /∈ S, Λa(s, µ) and Λb(s, µ) have analytic continuation to the wholes-plane, are bounded in every vertical strips, and satisfy the functional equation

(1.5.25) Λa(s, µ) = ikµ(N)χ(D)τ(µ)2

D(D2N)−s+

k2 Λb(k − s, µ)

Then f(z) =∑∞

n=0 anqn is a modular form of level N with nebentypus χ i.e. f ∈Mk(N,χ).

1.6. Hecke operators and Euler product. It is sometimes convenient to express theautomorphy condition (1.3.1) in terms of group action. For every matrix with positivedeterminant

γ =

[a bc d

]∈ GL2(R)+,

we define the right action of γ on a function f : H→ C by the transformation rule

(1.6.1) (f |k γ)(z) = det(γ)k/2(cz + d)−kf

(az + b

cz + d

)A straightforward calculation shows that (f |k α) |k β = f |k (αβ). Thus this transformationrule is indeed a right action of GL+

2 (R) on the space of holomorphic functions on H.With this definition, a modular form of weight k with respect to a Fuschian group Γ is a

holomorphic function f on H such that f |k γ = f for all γ ∈ Γ and which satisfies a growthcondition near the cusps. It also follows from this definition that the algebra of double cosetsof Γ in GL+

2 (Q) acts naturally on the space Mk(Γ) as well as Sk(Γ).The construction of the algebra of double cosets of a congruence subgroup Γ in Σ =

GL+2 (Q) relies on the following property.

Lemma 1.6.1. Let Γ be a congruence subgroup and α ∈ Σ. Then Γ ∩ αΓα−1 is a subgroupof Γ with finite index.

This lemma can be reformulated in the following way : Each double coset ΓαΓ in Σ is afinite union of right cosets or left cosets

(1.6.2) ΓαΓ =⊔i

Γαi

where i runs over a finite set of indexes. It is convenient to see double cosets as notion ofrelative position between two let cosets. We will say that the two cosets Γα1 and Γα2 are inposition α if α1α

−12 ∈ ΓαΓ. Lemma 1.6.1 can be reformulated in an yet another way :

Lemma 1.6.2. For each left coset Γα1 and each double coset ΓαΓ, there are only finitelymany left cosets Γα2 so that Γα1 and Γα2 are in position ΓαΓ.

Let HΓ denote the free abelian group generated by a basis indexed by the set of doublecosets of Γ in Σ. We simple write Tα ∈ HΓ for the element in this basis indexed by the doublcoset ΓαΓ. We define a multiplication on HΓ by the following rule

(1.6.3) Tα1Tα2 =∑α

cαα1,α2Tα

where cαα1,α2is the number of left coset Γβ such that Γβ1 and Γβ are in position α1 and Γβ

and Γβ2 are in position α2, here Γβ1 and Γβ2 are fixed cosets in position α. The finitenessof the numbers cαα1,α2

follows immediately from 1.6.2.18

We define the action of ΓαΓ =⊔i Γαi on Mk(Γ) by the formula

(1.6.4) f |k Tα =∑i

f |k αi.

This formula defines an action of the Hecke algebraHΓ on the space of modular forms Mk(Γ).This action preserves the subspace of cusp forms Sk(Γ).

We consider the Hecke algebra HΓ(1) with respect to the full modular group Γ(1) = SL2(Z)as subgroup of Σ = GL2(Q)+. The double cosets of Γ(1) be described explicitly by the theoryof elementary divisors.

Proposition 1.6.3. Each double coset of Γ(1) in GL2(Q)+ contains a unique diagonal matrixof the form

(1.6.5) α(d1, d2) =

[d1 00 d2

]with d1, d2 ∈ Q+ such that d1d

−12 is an integer.

Proof. This can be best explained in term of relative position between lattices. A lattice ofQ2 is a free abelian group of rank two contained in Q2. The map α 7→ α−1(Z2) defines abijection between the set of left coset Γ(1)α in Σ = GL+

2 (Q) into the set of lattices of Q2. IfL,L′ ∈ Q2 are two lattices, the theorem of elementary divisors assert that there exist a basisx1, x2 of L such that d1x1, d2x2 is a basis of L′ where d1, d2 are well defined positiverational numbers such that d1d

−12 is an integer.

Proposition 1.6.4. The algebra HΓ(1) is commutative.

Proof. The transposed matrix g 7→ g> being an anti-homomorphism of Σ, it induces ananti-homomorphism on HΓ(1). On the other hand, as it fixed the diagonal matrices α(d1, d2),it induces identity on HΓ(1). This means that HΓ(1) is a commutative algebra.

The Hecke operators are self-adjoint with respect to the Peterson inner product on SK(Γ(1)).Recall that this inner product is defined by the integral

(1.6.6) 〈f, g〉 =

∫Γ(1)\H

f(z)g(z)ykdxdy

y2.

Here the invariant dxdy/y2 on H and defines a measure on the quotient Γ(1)\H, the expres-sion f(z)g(z)yk is also invariant under Γ(1) and defines a function on Γ\H. Since f, g arecusp forms, f(z)g(z)yk tends to zero near the cusps and thus is bounded function on Γ\H.Now the Peterson inner product is well defined since Γ(1)\H has finite measure with respectto dxdy/y2.

Theorem 1.6.5. The action of HΓ(1) on Sk(Γ(1)) can be simultaneously diagonalized.

Proof. The Hecke operators are self-adjoint with respect to to the Peterson inner product.A commutative algebra of self-adjoint operators on a finite dimensional vector space can besimultaneously diagonalized.

We will next single out a particular family of Hecke operators that are relevant to L-functions. For every n ∈ N, we define the Hecke operator T (n) ∈ HΓ(1) by the following

19

formula

(1.6.7) T (n) =∑

d1,d2∈N,d2|d1d1d2=N

Tα(d1,d2).

The corresponding union of double cosets

(1.6.8) T (n) =⊔

d1,d2∈N,d2|d1d1d2=N

Γ(1)α(d1, d2)Γ(1)

is the set of integral matrices of determinant n

(1.6.9) T (n) = α ∈ Mat2(Z) | det(α) = n.

Proposition 1.6.6. Let f =∑∞

m=1 amqm be a cusp form of weight k for Γ(1). Let f |k

T (n) =∑∞

m=1 bmqm be the Fourier development of T (n)f . We have

(1.6.10) bm =∑

ad=n,a|m

nk2 d−k+1amd

a.

In particular

(1.6.11) b1 = n−k2

+1an.

Proof. We can make explicit the action of T (n) on modular forms by decomposing T (n) inleft cosets of Γ(1)

(1.6.12) T (n) =⊔

a,b,d∈Nad=n,0≤b<d

Γ(1)

[a b0 d

].

Thus

f |k T (n) =∑ad=n

∑0≤b<d

nk/2dkf

(az + b

d

)(1.6.13)

=∑ad=n

nk/2dk∞∑m=1

am e2πimaz

d

∑0≤b<d

e2πimbd(1.6.14)

The term∑

0≤b<d e2πimbd vanishes unless d|m in which case it is equal to d. By replacing m

by dm in the above formula, we get the following expression

f |k T (n) =∑ad=n

nk/2dk+1

∞∑m=1

amd e2πimaz(1.6.15)

=∞∑m=1

∑ad=n,a|m

nk/2dk+1amdaem(1.6.16)

from which we derive (1.6.10)

Corollary 1.6.7. The Fourier coefficient an of a normalized eigenform f of weight k with

respect to Γ(1) is the eigenvalue of the operator nk2−1T (n).

20

Theorem 1.6.8. Let f =∑∞

n=1 anqn be a cuspidal eigenform of weight k for Γ(1) normalized

so that a1 = 1. Then we have

(1.6.17) T (n)f = n−k2

+1anf.

In particular, we have the relation

(1.6.18) amn = aman

for all relatively prime integers m,n ∈ N. Moreover, the Dirichlet series L(s, f) admits adevelopment in Euler product

(1.6.19) L(s, f) =∞∑n=1

ann−s =

∏p

(1− app−s + pk−1−2s)−1.

Proof. Let c(n) denote the eigenvalue of T (n) with respect to the eigenvector f . We haveb1 = c(n)a1 = c(n) since a1 = 1 with our normalization. Now (1.6.17) follows from (1.6.11).

The multiplicative relation (1.6.18) follows from the relation in the Hecke algebra T (mn) =T (m)T (n) that holds for (m,n) = 1.

The multiplicative relation implies a development in product of the Dirichlet series

(1.6.20) L(s, f) =∞∑n=1

ann−s =

∏p

(∞∑r=0

aprp−rs).

The formula

(1.6.21)∞∑r=0

aprp−rs = (1− app−s + pk−1−2s)−1

follows from similar formula in the Hecke algebra.

1.7. Old and new forms. The theory of Hecke operators and expansion L-function as anEuler product can be generalized to any congruence subgroup. It will take however a muchmore complicated form. As the theory of Hecke operators can be significantly streamlinedwith the introduction of the adeles and the interpretation of modular form as automorphicforms on an adelic group, we will postpone discussion after the adeles being introduced.

For the record, we will just state the result of the theory of new forms, due to Atkin andLehner. The proof will be postponed.

Let M,N ∈ N such that M |N . For any character χ : (Z/MZ)× → C, we have a character(Z/NZ)× → C also denoted by χ obtained by composing with (Z/NZ)× → (Z/MZ)×. Forany integer d such that dM |N , there is a map

[d] : Sk(M,χ)→ Sk(N,χ)

that associate to a form f ∈ Sk(M,χ) the form z 7→ f(dz) that belongs to Sk(N,χ). Letdenote Sk(N,χ)old the subspace generated by the image of [d] for all integers d,M so thatdM |N . The orthogonal complement of Sk(N,χ)old is denoted Sk(N,χ)new.

Theorem 1.7.1. (1) The space of new forms Sk(N,χ)new admits a basis consisting ofnormalized eigenform for the operators Tp associated to the primes p not dividing N .

(2) The L-function associated to such a normalized eigenform has a complete expansionas a Euler product.

21

(3) If f, f ′ ∈ Sk(N,χ)new such that for every prime p not dividing N , there exists ap suchthat Tpf = apf and Tpf

′ = apf′ then f and f ′ are proportional.

2. Representations of GL(2,R)+

2.1. Representations of locally compact groups. Let G be a topological group that islocally compact. We will be interested in representations of G on Hilbert spaces.

Definition 2.1.1. A representation of G on a Hilbert space H is a homomorphism π fromG to the group of continuous linear transformations of H so that for evry v ∈ H, the mapg → π(g)v is continuous. If moreover π preserves the inner product on H, we will say thatthe representation π is unitary.

Lemma 2.1.2. Every representation is locally bounded. In other words, for every compactK of G, there exists a positive real number C such that |π(g)| < C for every g ∈ K.

Proof. For every v, the vectors π(g)v with g ∈ K form a compact set of H which is necessarilybounded. The lemma follows from the uniform boundedness principle.

Let Cc(G) denote the space of complex valued continuous functions on G with compactsupport. Assume that G is unimodular. The Haar measure dg defines a positive linear formCc(G)→ C

φ 7→∫G

φ(g)dg.

The Haar measure also provides Cc(G) with a structure of algebra under the convolutionproduct

(2.1.1) φ ∗ ψ(x) =

∫G

φ(xy−1)ψ(y)dy.

Let π be a representation of a locally compact topological group G on a Hilbert space g.We consider the integral

π(φ)v =

∫G

φ(g)π(g)vdg

for every φ ∈ Cc(G) and v ∈ H. The above integral can be defined as follows. We firstconsider the continous linear form

v′ 7→∫G

φ(g)〈π(g)v, v′〉dg.

By Riesz representation theorem there exists a unique vector π(φ)v ∈ H so that

〈π(φ)v, v′〉 =

∫G

φ(g)〈π(g)v, v′〉dg.

This defines an action of Cc(G) on H.

Definition 2.1.3. A sequence of positive functions φn is said to be approximating the deltafunction of the identity of G in the following sense

(1) φn ∈ Cc(G) are supported in a certain compact K of G(2)

∫Gφn(g)dg = 1 for every n

(3) for every neighborhood U of 1G we have limn→∞∫G−U φn(g)dg = 0.

22

Lemma 2.1.4. Let π be a representation of G on a Hilbert space H. If φn is a sequence offunction approximating the delta function of 1G, then we have

limn→∞

π(φn)v = v

for all v ∈ H.

Proof. Let v ∈ V . For every ε > 0, there exists a neighborhood U of 1G so that for everyg ∈ U , ||π(g)v−v|| < ε. For an integer n large enough, we have limn→∞

∫G−U ||φn(g)v||dg < ε

since the family |φn| is bounded and limn→∞∫G−U φn(g)dg = 0. We can also check that∫

U||π(φn)v − v|| < ε for n large enough.

2.2. Representations of the circle group. The theory of unitary representations of thecircle group is very much a reformulation of the series of Fourier series associated to squareintegrable functions on the circle T = R/Z. The characters of is of the forms

χn(x) = e2πinx.

Let H = L2(T) denote the space of square integrable functions on T. The translation by Tdefines a unitary representation of T on H.

Theorem 2.2.1. Let π be a unitary representation of T on a Hilbert space H. For everyv ∈ H and n ∈ Z, we consider

vn = π(χ−n)v =

∫Te−2πinxπ(x)vdx.

Let Hn denote the image pn : v 7→ vn = π(χ−n)v.

(1) For n 6= m, the subspaces Hn and Hm are orthorgonal.(2) The space H is the Hilbert direct sum of its subspace Hn.

Proof. Only the last statement is non trivial. For the last statement, it suffices to prove thatfor all v ∈ H, v = limN→∞

∑Nn=−N pn(v). Since v−

∑n=Nn=−N pn(v) is orthogonal to

⊕Nn=−N Hn,

for every v′ ∈⊕N

n=−N Hn, we have

|v − v′| ≥ |v −n=N∑n=−N

pn(v)|.

It follows that we only need to construct a sequence v′N ∈⊕N

n=−N Hn so that

limN→∞

|v − v′N | = 0.

This can be achieved by constructing a delta sequence of functions φN on T so that for allN , φN is a linear combination of e2πinx with |n| ≤ N . Classical examples of this is Fejer’skernel

KN(x) =N∑

n=−N

(1− |n|N + 1

)e2πinx(2.2.1)

=1

N + 1

sin2(π(N + 1)x)

sin2(πx)(2.2.2)

23

On the complement of neighborhood of 0 ∈ T, sin2(πx) > α for some given positive numberα which implies that KN(x) < e−1/(N + 1) for all N . This implies that Fejer’s sequenceapproximate the delta function.

A vector v of some finite direct sum⊕N

n=−N Hn is called a finite vector i.e. its transformsπ(x)v with x ∈ T generates a finite dimensional vector space. It follows from the abovetheorem that there are non zero finite vectors in any unitary representation of the circlegroups, and moreover, the finite vectors form a dense subspace of the Hilbert space. Thisstatement can be generalized to arbitrary compact group.

Lemma 2.2.2. Let (π,H) be a Hilbert representation of a compact group K.Then there existsa hermitian inner product on H inducing the same topology as the original one so that π isunitary i.e. 〈π(g)v, π(g)v′〉 = 〈v, v′〉 for all g ∈ K and v, v′ ∈ H.

Proof. [2, 2.4.3] Let 〈v, v′〉1 denote the given inner product on H. For all v, the function k 7→〈π(k)v, π(k)v′〉1 is a continuous function onthe compact group K which is then necessarilybounded. By the uniform boundedness principle, there exist a real number C > 0 such that|π(k)v| < C|v| for all non zero vector v ∈ H. This also implies that |π(k)v| > C−1|v|.

The inner form

〈v, v′〉 =

∫K

〈π(k)v, π(k)v′〉1dk

is obviously positively definite. The inequalities C−1|v| < |π(k)v| < C|v| imply that itdefines the same topology on H as 〈v, v′〉1.

2.3. Lie groups. Let G be a real Lie group. Let g denote its Lie algebra.

Proposition 2.3.1. There exists natural isomorphism between

(1) The tangent space of G at the origin g.(2) The space of left invariant vector fields on G.(3) The space of left invariant derivations of C∞(G).

For every vector X ∈ g, there is a unique invariant vector field LX having X as thevector X at the origin. For every g ∈ G, the left translation map lg : G → G induces anisomorphism dlg : g → TgG and we have LX,g = dlg(X). Left invariant vector fields areequivalent to left invariant derivations. We have the following formula for the bracket

L[X,Y ]f = (LXLY − LYLX)f.

Proposition 2.3.2. There exists a map exp : g → G that is a local homeomorphism in aneighborhood of the origin in g such that, for any X ∈ g, t 7→ exp(tX) in an integral curvefor the left invariant vector field LX . Moreover exp((t+ u)X) = exp(tX) exp(uX).

For every smooth function f ∈ C∞(G), we have the following formula for the left invariantderivation

(LXf)(g) =d

dtf(g exp(tX))|t=0.

Definition 2.3.3. A representation of the Lie algebra is a linear application π : g→ End(V )from g to the space of endomorphisms of a vector space V such that for all X, Y ∈ g , wehave

π[X, Y ] = π(X)π(Y )− π(Y )π(X).24

Let us consider examples of finite dimensional representation of Lie algebras. Let G =SL2(R) and g = sl2(R) its Lie algebra. Let V be the standard 2-dimensional vector space onwhich G acts. For every integer k, Symk−1(V ) is a vector space of dimension k equipped withan induced action of G. This representation is algebraic and its derivation is a representationof Lie algebras g → gl(Symk−1(V )). But we are mostly interested in infinite dimensionalrepresentations.

We define the universal enveloping algebra U(g) as the quotient of the tensorial algebra⊗g =

⊕N

⊗n g by the ideal generated by [x, y] − x ⊗ y + y ⊗ x. Every representationπ : g→ End(V ) can be extended in a unique way into a homomorphism of algebras U(g)→End(V ).

The Killing form is the symmetric bilinear form on g defined by

B(x, y) = Tr(ad(x)ad(y)).

Since it is invariant under the adjoint action of G on g, after derivation we get

B([z, x], y) +B(x, [z, y]) = 0.

Proposition 2.3.4. Assume that the Killing form is non degenerate. Let x1, . . . , xd be a basisof g and y1, . . . , yd be the dual basis so that B(xi, yj) = δij. Then the element Ω =

∑i xiyi

does not depend on the choice of the basis. Moreover, it belongs to the center of U(g).

Proof. The adjoint action of G on g induces an action of G on g. Its derivation is an actionof g on U(g). We can check that this action is X(A) = XA−AX. Sice Ω is fixed under theaction of G, it follows that it commutes with g. Since g generates U(g), Ω belongs to thecenter of U(g).

Let us consider the case g = sl2(R). Its standard base

(2.3.1) H+ =

[1 00 −1

], R+ =

[0 10 0

], L+ =

[0 01 0

]satisfies the commutation rule [H+, R+] = 2R+, [H+, L+] = −2L+ and [R+, L+] = H+. Thedual basis with respect to the Killing form is H+, 2L+, 2R+. The Casimir element is

(2.3.2) Ω = H2+ + 2R+L+ + 2L+R+

It belongs to the center of the universal enveloping algebras. In fact the Casimir element isa generator of the center of the universal algebra of sl2.

Proposition 2.3.5. The Casimir element Ω acts on the irreducible n-dimensional represen-tation of g by the scalar n2 − 1.

2.4. Smooth, analytic and K-finite vectors in a Hilbert representation. A repre-sentation of a Lie group G on a topological vector space H is a continuous linear actionπ : G× H→ H. If H is a Hilbert space, we call π a Hilbert representation. Observe that wedo not require that the action of G preserve the inner product of H. If it does, we say thatπ is a unitary representation.

A vector v ∈ H is C1 if for every X ∈ g, the limit

π(X)v := limt→0

1

t(π(exp(tX))v − v)

25

exists. Recursively, we can define Ck-vectors for all k ∈ N. A smooth or C∞-vector v is avector that is Ck for all k. The space of smooth vectors is a representation of the Lie algebrag.

Proposition 2.4.1. Let π be a Hilbert representation of a Lie group G in a Hilbert spaceH. Then the subspace of smooth vectors Hsm is dense in H.

Proof. For a continuous function with compact support φ ∈ Cc(G), we define

π(φ)v =

∫G

φ(g)π(g)v dg.

If φ is a smooth function, π(φ)v is a smooth vector. The proposition follows from theexistence of a delta sequence of smooth compactly supported function.

Proposition 2.4.2. Let (π,H) be an irreducible unitary representation of G and Hsm thedense subspace of smooth vectors. Then all elements Ω ∈ Z(U(g)) acts on Hsm as a scalar.

Proof. This is an elaborate version of the Schur lemma. See [11, 1.6.5].

We can assume that the restriction of H to K is unitary. Then we have a decompositionof H as Hilbert direct sum

H =⊕

`∈ZH(`).

The algebraic direct sum⊕

`∈Z H(`) is the subspace of K-finite vectors.

Proposition 2.4.3. For every ` ∈ Z, the subspace Hsm∩H(`) is dense in H(`). The subspaceof smooth, K-finite vectors is dense in H.

Proof. [11, 3.3.5].

We will also consider the subspace Han of analytic vectors. A vector v ∈ H is analyticif the function g 7→ π(g)v is analytic. It is equivalent to the apparently weaker conditionthat for all w ∈ H, the function g 7→ 〈π(g)v, w〉 is analytic [12, p.278]. An analytic vectoris obviously a smooth vector. The space Han is a subspace of Hsm which is table under theaction of g.

Proposition 2.4.4. Smooth, K-finite vectors are analytic.

Proof. In order to prove that a function is analytic, it is enough to prove that it is annihilatedby an elliptic differential operator. Recall that the Casimir element

Ω = H2+ + 2R+L+ + L+R+ = H2 + (R+ + L+)2 − (R+ − L+)2

acts as scalar on smooth vectors. If v is K-eigenvector then it is also an eigenvector of(R+ −L+). It follows that it is an eigenvector of the elliptic operator Ω + 2(R+ −L+)2.

Lemma 2.4.5. Let G be a connected Lie group. If V is a g-invariant subspace of Han, thenits closure V is a G-invariant subspace of H.

Proof. See [11, 1.6.6]. If V ⊥ denote the orthogonal complement of V , then V = (V ⊥)⊥. LetX ∈ g, v ∈ V and w ∈ V ⊥. There exists ε > 0 such that if |t| < ε then

〈π(exp(tX))v, w〉 =∑n

tn

n!〈π(X)nv, w〉

26

and the series converges absolutely.Since π(X)nv ∈ V and w ∈ V • it follows that 〈π(exp(tX))v, w〉 = 0 for |t| < ε. The real

analiticity of the function t 7→ 〈π(exp(tX))v, w〉 implies that 〈π(exp(X))v, w〉 = 0 for allv ∈ V , w ∈ W and X ∈ g. It follows that V is stable under exp(X) for all X ∈ g thus underG since exp(g) generates G.

Let G be a real reductive Lie group, K a maximal compact subgroup of G anf g its Liealgebra. By a (g, K)-module, we mean a vector space V together with representation π ofK and of g subject to the following conditions

(1) V is a direct sum of finite dimensional irreducible representations of K(2) The actions of K and of g are compatible i.e. for every X in the Lie algebra of K

and for every vector v ∈ V , we have the relation

(2.4.1) π(X)v =d

dt(exp(tX))f |t=0.

(3) For every g ∈ K and X ∈ g, we have the relation

(2.4.2) π(g)π(X)π(g−1)v = π(Ad(g)X)v.

The module is said to be admissible if each irreducible representations of K occurs withfinite multiplicity.

Proposition 2.4.6. Let (π,H) be an irreducible Hilbert representation of G. Then thesubspace of smooth K-finite vectors V is a (g, K)-module.

Proof. The commutation relation (2.4.1) and (2.4.2) can be checked upon the definition ofπ(X). If v is a K-finite vector, then so is π(X)v because the finite dimensional vector spacegenerated by π(Y )π(k)v with Y ∈ g and k ∈ K is stable under the action of K. As in 2.2.2,the Hermitian form can be made K-invariant without changing the underlying topology ofH.

Proposition 2.4.7. Let V be a finitely generated (g, K)-module on which Ω acts as a scalar.Then V is admissible.

Proposition 2.4.8. Let V be an irreducible (g, K)-module. Then the Casimir element Ωacts on it as a scalar.

Proof. We first prove that V as a countable basis. For every v ∈ V , the vectors kv withk ∈ K span a finite dimensional subspace span(Kv). The space U(g)span(Kv) is then anonzero (g, K)-submodule of V . Since V is irreducible, we have V = U(g)span(Kv) and inparticular, V has a countable basis.

The following version of Schur lemma, due to Dixmier, implies that Ω acts as a scalar. Wewill see later that from this fact, we can classify completely all irreducible (g, K)-modulesand then derive their admissibility.

Lemma 2.4.9. Suppose that V is countable dimensional and that S ⊂ End(V ) act irre-ducibly. If T ∈ End(V ) commutes with all elements of S then T is a scalar multiple ofidentity.

Proof. It is enough to prove that there exists x ∈ C so that T − c1V is not invertible. If itis the case, either its kernel or its image is a proper subspace which is stable under S. Thiswould contradict the irreducibility.

27

If for all c ∈ C, T − c1v is irreducible. It follows that for all polynomial q(t) ∈ C[t], Q[T ] isinvertible and therefore we can define the operator R[T ] for all rational fraction R(t) ∈ C(t).We derive a linear injective map C(t) → V defined by r(t) 7→ r[T ]v. This implies C(t) hasa countable basis which is a contradiction.

Proposition 2.4.10. Let (π,H) be a irreducible unitary representation of G. Then V =Hsm ∩ Hfin is an irreducible admissible (g, K)-module.

Proof. Let v ∈ V a non zero vector. Then V ′ = U(g)span(Kv) is a (g, K)-module of finitetype on which Ω acts as a scalar. It follows that its is admissible i.e for every ` ∈ Z, V ′(`)is finite dimensional. But we know that V ′ is a g-invariant subspace of Han, then V ′ is

G-stable thus V ′ = H. It follows that H =⊕

`∈ZV′(`). It follows that V (`) = V ′(`) is finite

dimensional and H is admissible.

Two Hilbert representations (π,H) and (π′,H′) of G are said to be infinitesimally equiva-lent if their associated (g, K)-modules are isomorphic. A priori there exist non isomorphicirreducible Hilbert representations that are infinitesimally equivalent. But as we will see inthe case of GL2(R), irreducible unitary representations that are infinitesimally equivalent,have to be isomorphic.

2.5. (g, K)-modules for GL2(R). Let G = GL2(R)+ and K = SO2(R) its maximal compactgroup.

K =

[cos θ sin θ− sin θ cos θ

]| θ ∈ R/2πZ

.

Since K is isomorphic to the circle group R/Z, all irreducible unitary representations of K areone dimensional and classified by the integers. For an integer ` ∈ Z, the `-th representationof K is given by the `-th power in the circle group.

Let V be admissible (g, K)-modules. We have a decomposition of V into algebraic directsum

V =⊕`∈Z

V (`)

where K acts on V (`) by its `-th power.In sl2(C), we consider the following triple

(2.5.1) H = −i[

0 1−1 0

], R =

1

2

[1 ii −1

], L =

1

2

[1 −i−i −1

]which is obtained from the triple (H+, R+, L+) by conjugation by the Cayley matrix

(2.5.2) c = −i+ 1

2

[i 1i −1

]i.e. c−1(H+, R+, L+)c = (H,R,L). We also have

(2.5.3) c−1

[eiθ 00 e−iθ

]c =

[cos θ sin θ− sin θ cos θ

].

Thus H is a generator of the complexified Lie algebra of K. In fact, the Lie algebra of circlegroup K is generated by the vector

(2.5.4) W =

[0 1−1 0

]28

and we have H = −iW .Since Ω is invariant under the adjoint action, we have

(2.5.5) Ω = H2 + 2RL+ 2LR.

By Dixmier’s lemma 2.4.9, Ω acts on an irreducible (g, K)-module by a scalar.

Proposition 2.5.1. Let V be admissible (g, K)-modules and V =⊕

` V (`) be is decomposi-tion in direct sum of isotypical K-components.

(1) V (`) is the space of vectors v ∈ V such that Hv = kv.(2) If v ∈ V (`), then Rv ∈ V (`+ 2) and Lv ∈ V (`− 2). Moreover, if v 6= 0, then Rnv is

a generator of V (`+ 2n) and Lnv is a generator of V (`− 2n).(3) The dimension of V (`) is at most one. If V (`) 6= 0 and V (k) 6= 0 then k − l is an

even integer. If V (`) 6= 0 for some even (resp. odd) integer, we say that V is an even(resp. odd) module.

(4) Let Ω act on V by the scalar ω. For v ∈ V (`), we have

(2.5.6) 4LRv = (ω − `(`+ 2))v , 4RLv = (ω + `(2− `))v.If v 6= 0 and Rv = 0 then ω = (`+1)2−1. If v 6= 0 and Lv = 0 then ω = (`−1)2−1.

Proof. For v ∈ V (`), we have

Wv =d

dtei`tv|t=0 = i`v.

It follows that Hv = kv.By using the commutation rule [H,R] = 2R, we have

HRv = RHv + [H,R]v = (`+ 2)Rv

that implies Rv ∈ V (`+ 2). We have similarly HLv = (`− 2)Lv.By using the commutation rule [R,L] = 2H, we have

4LRv = Ωv −H2v − 2[R,L]v = (ω − `(`+ 2))v.

The other affirmations follow from these computations.

Definition 2.5.2. Let V be an irreducible admissible (g, K)-module. The K-type of V isthe set Σ(V ) of integers k ∈ Z so that V (`) 6= 0. The central character of V is given by theeigenvalue α of Z. The infinitesimal character is given be the eigenvalue ω of Ω.

Corollary 2.5.3. The isomorphism class of an irreducible admissible (g, K)-module is de-termined by its central character, its infinitesimal character and its K-type Σ(V ). Let ω bethe eigenvalue of Ω on V (the infinitesimal character). Let us write ω = s2 − 1 with s ∈ Cwell determined up to a sign. Let ε ∈ 0, 1 be the parity of V :

(1) If s in not an integer, or an integer congruent to ε modulo 2 then the K-type of V is

Σ = ` ∈ Z | ` ≡ ε mod 2.(2) If s is an integer not congruent to ε modulo 2, then the K-type of V can be either

Σ+ = ` ≥ |s|+ 1 | ` ≡ ε mod 2Σ0 = −|s| − 1 < l < |s|+ 1 | ` ≡ ε mod 2Σ− = l ≤ −|s| − 1 | ` ≡ ε mod 2

Note that the case s = 0 is exceptional because in that case Σ0 = ∅.29

In every case, there exists nonzero vector v` ∈ V (l) with l in the set of K-type a a goodchoice of s so that

Hv` = lv`

Rv` =s+ 1 + `

2v`+2

Lv` =s+ 1− `

2v`−2

In the first case, the choice of s between the two solutions of teh equation ω = s2 − 1 doesnot matter. In the second case, we have to choose positive s in the cases Σ± and negative sin the case Σ0.

2.6. Unitary (g, K)-modules. Let (π,H) be a unitary representation of G. The space ofK-finite vectors V = Hfin is then a dense subspace of G which is equipped with a structureof (g, K)-module.

Proposition 2.6.1. If V = Hfin is the (g, K)-module associated with a unitary representationthen the action of K is unitary and the action of fg is anti-self-adjoint i.e. that for all X ∈ g,v, w ∈ V , we have

〈Xv,w〉 = −〈v,Xw〉.

Proof. We derive 〈Xv,w〉+ 〈v,Xw〉 = 0 from the relation

〈exp(tX)v, exp(tX)w〉 = 〈v, w〉by Leibnitz rule.

Recall that R = 12(H+ + iX) and L = 1

2(H+ − iX) where H+, X ∈ g are real matrices. In

other words, R,L are complex conjugate matrices. It follows that

〈Rv,w〉 = −〈v, Lw〉for all v, w ∈ V .

Proposition 2.6.2. If (π1,H1) and (π2,H2) are irreducible unitary representations of G,they are unitarily equivalent if and only if they are infinitesimally equivalent i.e. if V1 = Hfin

1

and V2 = Hfin2 are isomorphic as (g, K)-modules.

Proof. Let k ∈ Z so that V1(`) 6= 0. Then V2(`) 6= 0. Choose non zero vector x1 ∈ V1(`) andx2 ∈ V2(`) so that 〈x1, x1〉 = 〈x2, x2〉. It is enough to prove 〈Lx1, Lx1〉 = 〈Lx2, Lx2〉 and thesame for R because Lnx1 and Rnx1 form a basis for V1. But this follows from the calculation

〈Lx1, Lx1〉 = −〈RLx1, x1〉(2.6.1)

=1

4(ω − `(`+ 2))〈x1, x1〉(2.6.2)

and the identical calculation for x2.

Proposition 2.6.3. Let V be an infinite dimensional irreducible unitary (g, K)-module.Then the central element Z act on V by a purely imaginary scalar µ and the Casimir elementΩ act by a real scalar ω. Moreover, if we write ω in the form ω = s2− 1 then there are onlyfour possibilities

(1) s is purely imaginary (unitary principal series),30

(2) s is real and −1 < s < 1 (complementary series),(3) s ∈ Z (discrete series and limit of discrete series),

Proof. Since V is irreducible Z and Ω ought act by scalars. Since Z is skew-symmetric,the scalar µ has to be a purely imaginary number. Since Ω = H2

+ + 2R+L+ + 2L+R+ issymmetric, the scalar ω has to be a real number.

Assume that ω is not of the form s2 − 1 for an integer s, then ω < 0 (resp. ω < −1)if V is an even (resp. odd) module. In fact, if ω is not of the form s2 − 1 with integer sthen V (`) 6= 0 for all even (resp. odd) integer k depending on the parity of V . Moreover, ifv ∈ V (`) is a non zero vector, then Rv 6= 0. It follows from

(ω − `(`+ 2))〈v, v〉 = 〈v, 4LRv〉 = −4〈Rv,Rv〉 < 0

that ω − `(`+ 2) < 0 for all even (resp. odd) integer k if V is even (resp. odd). In the firstcase, we have ω < 0 by taking ` = 0. In the second case, we have ω < −1 by taking ` = −1.

• If ω < −1, then s2 is a real negative integer i.e. s is purely imaginary.• If −1 ≤ ω < 0, then s is a real integer in the interval (−1, 1). In this case V must be

an even module.• The remaining case ω = s2 − 1 with s ∈ N. If s ≥ 1, V is called a discrete series. Ifs = 0 then V is called limit of discrete series.

In the subsequent paragraphs, we will construct unitary representations of G who (g, K)-modules fit with the three cases enumerated above.

In order to complete the picture, we will have to construct unitary representations in threeclasses enumerated in Proposition 2.6.3 : unitary principal series, complementary series anddiscrete series. For later references, we we will name

• Ps the (g, K)-module in the unitary principal series, associated to a purely imaginarynumber ±s,• Cs the (g, K)-module in the complementary principal series, associated to a purely

real number −1 < ±s < 1,• D±k the (g, K)-module in the discrete series. A (g, K)-module with a lowest weightk ∈ N is denoted Dk, in this case s = ±(k−1). A (g, K)-module with a lowest weight−k ∈ −N is denoted D−k, in this case s = ±(k − 1).

2.7. Unitary principal series of GL2(R)+. Let B denote the subgroup of triangular ma-trices of GL2(R)

(2.7.1) b =

[y1 x0 y2

]=

[y1 00 y2

] [1 y−1

1 x0 1

]=

[1 y−1

2 x0 1

] [y1 00 y2

]with y1, y2 ∈ R×, x ∈ R. On B, the left and right invariant measures can be expressed asfollows

dl(B)b =dy1dy2dx

|y1|2|y2|and dr(B)b =

dy1dy2dx

|y1||y2|2.

As δB(b)dl(B)b = dr(B)b, the modulus character δ is given by the expression

δB(b) = |y1||y2|−1.

For all s1, s2 ∈ C and ε : ±1 → C×, we have a character χ : B → C×

χ(b) = ε(sgn(y1))|y1|s1 |y2|s231

which is unitary if and only if both s1, s2 are purely imaginary. For every character χ of B,let H0(χ) be the space of continuous functions f : G→ C so that

f(bg) = δB(b)1/2χ(b)f(g)

We have a representation of G on the Banach space H0(χ) equipped with the ∞-norm.

Lemma 2.7.1. Let H0(ε) denote the space of continuous functions f : K → C so thatf(−k) = ε(−1)f(k). If χ(b) = ε(sgn(y1))|y1|s1 |y2|s2 as above, the restriction to K defines alinear bijection H0(χ) → H0. The subspace of K-finite vectors of H(χ) is

⊕n even Hn if ε is

trivial and⊕

n odd Hn if ε is non trivial.

Proof. This follows from the Iwasawa decomposition G = BK and B ∩K = ±1.

Proposition 2.7.2. Let H(ε) be the space of L2-functions on K satisfying f(−k) = ε(−1)f(k).For every character χ(b) = ε(sgn(y1))|y1|s1|y2|s2, let H(χ) be the space of functions on G sat-isfying f(bg) = δB(b)1/2χ(b)f(g) so that the restriction to K is square integrable. Then therestriction to K defines an isomorphism H(χ) → H(ε). Moreover the right action of G onH(χ) defines Hilbert representation IndGB(χ) on the Hilbert space H(ε).

The inner product for every f1, f2 ∈ H(χ) is given by

(2.7.2)

∫K

f1f2(k)dk

so that the restriction of IndGB(χ) to K is unitary. But the representation IndGB(χ) of G isnot in general unitary.

Proposition 2.7.3. IndGB(χ) is unitary if χ is unitary.

Proof. For every f1, f2 the function f1f2(g) = f1(g)f2(g) satisfies the equation f1f2(bg) =δB(b)f1f2(g). As in C.2.1, there exists a canonical linear map

Cc(B\G, δB)→ Cthat is G invariant. Moreover, this linear form is proportional to the integration over K byC.2.2 so that the integral is G-invariant i.e. IndGB(χ) is unitary.

By Iwasawa decomposition G = BK, any matrix g ∈ GL2(R)+ can be written uniquelyunder the form

(2.7.3)

[a bc d

]=

[uy−1/2 0

0 uy−1/2

] [y x0 1

] [cos θ sin θ− sin θ cos θ

]with z, y ∈ R+, x ∈ R and θ ∈ R/2πZ. Note that we have written the central matrix in acomplicated way to make sure that det(g) = u2.

Proposition 2.7.4. For every χ = (ε, s1, s2), the space of K-finite vectors V (χ) in H(χ) isgenerated by the vectors f`

(2.7.4) f`

([uy−1/2 0

0 uy−1/2

] [y x0 1

] [cos θ sin θ− sin θ cos θ

])= us1+s2y

s+12 eil`θ

indexed by even (resp. odd) integers ` if ε is trivial (resp. non trivial). Here we haveintroduced the notation s = s1 − s2.

32

The action of g on Hfin is given by the formulas

(2.7.5) LHf` = lf`, LRf` =s+ 1 + `

2f`+2, LLf` =

s+ 1− `2

f`−2

and LZf` = (s1 + s2)f`. In particular, the Casimir element (2.5.5) acts as follows

Ωf` = (s2 − 1)f`.

The proposition from direct calculations of Lie derivatives. The following tabulations canbe found in [2, p.155] and [4, p.116] with slightly different coordinates. We follow Lang’scalculations.

Proposition 2.7.5. We have the following formula for the Lie derivatives LH ,LR,LL onC∞(G)

LH = −i ∂∂θ

(2.7.6)

LR = e2iθ

(iy∂

∂x+ y

∂

∂y− i

2

∂

∂θ

)(2.7.7)

LL = e−2iθ

(−iy ∂

∂x+ y

∂

∂y+i

2

∂

∂θ

)(2.7.8)

Proof. To all X ∈ g is attached a left invariant vector field whose associated flow is gX(t) =g exp(tX) on G. We have

φ(gX(t)) = φ(uX(t), xX(t), yX(t), θX(t))

and

(2.7.9) LXφ(g) =duX(t)

dt|t=0

∂φ

∂u+dxX(t)

dt|t=0

∂φ

∂x+dyX(t)

dt|t=0

∂φ

∂y+dθX(t)

dt|t=0

∂φ

∂θ.

We are thus about to calculate the derivatives dzX(t)dt|t=0,

dxX(t)dt|t=0,

dyX(t)dt|t=0 and dθX(t)

dt|t=0

for any X ∈ g.We need to remember the formula relating the coordinates (z, x, y, θ) with the matrix

entries a, b, c, d in the formula (2.7.3). Apply g to i ∈ H we get the equality

ai+ b

ci+ d= yi+ x

that allows us to express x, y as functions of a, b, c, d. We can also derive from the formula

(2.7.10)

[a bc d

]= uy−1/2

[y cos θ − x sin θ y sin θ + x cos θ− sin θ cos θ

]the equality d− ic = teiθ that allows us to calculate t as the modulus of d− ic and θ as itsargument. In particular we have

(2.7.11)d− ic|d− ic|

= eiθ.

Note that we also have the formula

(2.7.12) y =ad− bc|ci+ d|2

.

33

For R+ =

[0 10 0

], we have

exp(tR+) =

[1 t0 1

]It follows that

gR+(t) =

[a at+ bc ct+ d

]gR+(t)i =

ai+ at+ b

ci+ ct+ d= xR+(t) + iyR+(t)

= zR+(t)

We go on to calculate the derivative of x, y with respect to t

dzR+

dt|t=0 =

ad− bc(ci+ d)2

=ad− bc|ci+ d|2

(d− ci)2

|d− ci|2

= ye2iθ

after (2.7.11) and (2.7.12). In particular, we have the formulae

x′R+(0) = y cos 2θ

y′R+(0) = y sin 2θ

We calculate the derivative of θ

iθR+(t) = logct+ d− ic

((ct+ d)2 + c2)1/2

iθ′R+(0) =

(c2 + d2)1/2

d− icc(c2 + d2)1/2 − cd(d− ic)(c2 + d2)−1/2

c2 + d2

=c(d+ ic)

c2 + d2− cd

c2 + d2

=ic2

c2 + d2

θ′R+(t) =

c2

c2 + d2

= sin2 θ

The derivative of u vanishes because det(g) = u2 and det exp(tR+) = 1. From these calcu-lations we obtain

(2.7.13) LR+ = y cos 2θ∂

∂x+ y sin 2θ

∂

∂y+ sin2 θ

∂

∂θ.

For W =

[0 1−1 0

], we have

34

exp(tW ) =

[cos t sin t− sin t cos t

]g exp(tW ) =

[uy−1/2 0

0 uy−1/2

] [y x0 1

] [cos(θ + t) sin(θ + t)− sin(θ + t) cos(θ + t)

]and we get

(2.7.14) LW =∂

∂θ.

For H+ =

[1 00 −1

], we have

exp(tH+) =

[et 00 e−t

]g exp(tH+) =

[aet be−t

cet de−t

]g exp(tH+)i =

iaet + be−t

icet + de−t

= xH+(t) + iyH+(t)

iθH+(t) = logde−t − icet

(c2e2t + d2e−2t)1/2.

From this we obtain the Lie derivative

(2.7.15) LH+ = −2y sin 2θ∂

∂x+ 2y cos 2θ

∂

∂y+ sin 2θ

∂

∂θ.

We can now calculate the Lie derivatives LH ,LR,LL as linear combination with complexcoefficients of LR+ ,LW ,LH+ . We have

LH = −iLW

= −i ∂∂θ

LR = iLR+ −i

2LW +

1

2LH+

= e2iθ

(iy∂

∂x+ y

∂

∂y− i

2

∂

∂θ

)LL = −iLR+ +

i

2LW +

1

2LH+

= e−2iθ

(−iy ∂

∂x+ y

∂

∂y+i

2

∂

∂θ

)according to (2.5.1).

35

2.8. Complementary series of GL2(R)+. The construction of the unitary principal serieswas based on the fact that for purely imaginary numbers s1, s2, the induced representationH(s1, s2) affords a positively definite Hermitian inner form. In fact, for all copmlex numberss1, s2 ∈ C there is a Hermitian pairing

H∞(s1, s2)× H∞(−s1,−s2)→ C.

The construction of complementary series is based on the existence of an intertwining oper-ator.

H∞(s1, s2)→ H∞(s1, s2)

so that H∞(s1, s2) affords a Hermitian inner form if s1 = −s2 and s2 = s1. The lattercondition is equivalent with s1 + s2 ∈ iR and s = s1 − s2 ∈ R. We will construct theintertwining operator and prove that the induced inner form is positively definite is s =s1 − s2 ∈ (−1, 1).

For every smooth vector f ∈ H(χ) with χ = (ε, s1, s2), we consider the integral

(2.8.1) (M(s)f)(g) =

∫ ∞−∞

f

([0 −11 0

] [1 x0 1

]g

)dx.

Proposition 2.8.1. Suppose that <(s) > 0. For every K-finite vector f ∈ V (ε, s1, s2),the integral (2.8.1) is convergent, and M(s)f is a K-finite vector in H(ε, s2, s1). If f isK-finite, then so is M(s)f , and f 7→ M(s)f defines a homomorphism of (g, K)-modulesV (ε, s1, s2)→ V (ε, s2, s1).

Proof. Assume more generally that f is a smooth vector. Let us evaluate the integral M(s)fat g = 1. For that, we use Iwasawa decomposition

(2.8.2)

[0 −11 0

] [1 x0 1

]= (1 + x2)1/2

[(1 + x2)−1 −x(1 + x2)−1

0 1

]kθ(x)

where k ∈ K. It follows that

(2.8.3) M(s)f(1) =

∫ ∞−∞

(1 + x2)−1+s

2 f(kθ(x))dx.

If f is smooth, its restriction to K is bounded so that the above integral is absolutelyconvergent as long as <(s) > 0.

For every g ∈ G, the integral M(s)f(g) is still absolutely convergent since one can replacef by its right translated by g. In fact, the operator M(s) commutes with the right translationby G as long as the integrals converge. In particular, M(s) commutes with the right actionof K, and therefore it transforms a K-finite vector onto a K-finite vector.

We derive from the commutation rule

(2.8.4)

[0 −11 0

] [1 x0 1

] [y1 00 y2

]=

[y2 00 y1

] [0 −11 0

] [1 y−1

1 y2x0 1

]that if f ∈ V (ε, s1, s2) then M(s)f ∈ V (ε, s2, s1).

Proposition 2.8.2. Assume that <(s) > 0, then we have

(2.8.5) M(s)f`,s = (−i)`√π

Γ( s2)Γ( s+1

2)

Γ( s+1+`2

)Γ( s+1−`2

)f`,−s.

36

Proof. We already know that M(s)f`,s is a multiple of f`,−s so that it is enough to calculatethe constant M(s)f`,s(1). See [2, p.230] for the calculation of the constant.

Theorem 2.8.3. The intertwining operator M(s) : V (ε, s1, s2)→ V (ε, s2, s1) for <(s) > 0 bythe integral (2.8.1) can be meromorphically continued to the whole complex plane. It inducesa positively definite inner form on H(ε, s1, s2) for all real numbers s = s1 − s2 satisfying−1 < s < 1.

Proof. It follows from 2.8.2 that M(s) can be meromorphically continued to the whole com-plex plane.

For every complex numbers s1, s2, there is a Hermitian pairing

H∞(s1, s2)× H∞(−s1,−s2)→ C

defined by

(f1, f2) 7→∫K

f1(k)f2(k)dk.

By C.2.2, this pairing is G-invariant.Now we have a meromorphic family of intertwining operators

M(s) : V (s1, s2)→M(s2, s1).

If s1 + s2 is purely imaginary and s = s1 − s2 is real, then s1 = −s2 and s2 = −s1 so thatwe have a Hermitian inner product on H∞(ε, s1, s2).

It remains to check that for all even integers ` the constants

(−1)`/2√π

Γ( s2)Γ( s+1

2)

Γ( s+1+`2

)Γ( s+1−`2

)

are positive for all real numbers 0 < s < 1. Suppose ` ≥ 0. It suffices now to observe thatin this product Γ( s

2)Γ( s+1

2) and Γ( s+1+`

2) are positive, and the sign of Γ( s+1−`

2) is (−1)`/2.

Without this explicit formula we can check that the inner form is positive definite asfollows. If v is the generator of V (`) then we have

〈Rv,Rv〉 = (`(`+ 2)− ω)〈v, v〉

Here since ` is even integer `(`+ 2) ≥ 0. We also know that ω < 0. It follows that 〈Rv,Rv〉and 〈v, v〉 have the same sign.

2.9. Discrete series of GL2(R)+. We recall that on the upper half plan H, we have aninvariant measure µ = dxdy/y2. For every integer k ≥ 2, let µk = yk dxdy

y2 . We consider thevector space

(2.9.1) Hk = L2hol(H, µk)

of holomorphic functions on H which are square integrable with respect to the measure µk.The following proposition shows that Hk is complete with repect to the topology defined bythe L2-norm, and thus is a Hilbert space.

Lemma 2.9.1. Let fn be a sequence of holomorphic functions on the open disc D whichis L2-convergent. Then fn converges uniformly to a holomorphic function on any compactset.

37

Proof. [4, p.182] Recall the Cauchy formula for a holomorphic function f the unit disc D

f(0) =1

2π

∫ 2π

0

f(reiθ)

reiθdθ

thus

|f(0)| ≤ 1

2π

∫ 2π

0

|f(reiθ)|r

dθ.

It follows that

|f(0)|δ3

3≤ 1

2π

∫ δ

0

∫ 2π

0

|f(reiθ)|r

dθ.

It follows that |f(0)| and more generally the uniform norm is dominated by the local L1-normand therefore the local L2-norm.

Since the uniform norm is dominated by the L2-norm, a L2-convergent sequence fn isalso for the uniform topology. The limit is necessarily a holomorphic function again byapplication of the Cauchy formula.

Proposition 2.9.2. For every

g =

[a bc d

]∈ G

let us define

(πk(g−1)f)(z) = f |kg = (ad− bc)k/2(cz + d)−kf(gz).

Then πk is an irreducible unitary representation on Hd.

Proof. We have already seen that f 7→ f |kg defines a representation. We now verify thatπk(g

−1) preserves the L2-norm. If z′ = gz with real coordinates z′ = x′ + iy′, then we have

y′ =ad− bc|cz + d|2

y

according to (1.1.2) and µ′z = µz according to 1.1.2. Now, for every f ∈ Hk we have

||πk(g−1)f ||2 =

∫H

|f(z′)|2 (ad− bc)k

|cz + d|2kykµz

=

∫H

|f(z′)|2y′kµz′

= ||f ||2

which proves the unitarity of the representation πk.

With help of the Cayley transform, Hd is equipped with a convenient orthogonal basis.Recall that the Cayley map 1.1.3

z 7→ w =z − iz + i

defines an analytic isomorphism C between the upper half plane H and the open unit discD. Let us write w = u+ iv in cartesian coordinates and w = reiθ in polar coordinates.

38

Proposition 2.9.3. The isomorphism maps the density µk of H on the density νk of Hgiven by the formula

νk =4(1− |w|2)k−2dudv

|1− w|2k.

In particular, the Cayley map induces an isomorphism of Hilbert spaces

L2(H, µk) = L2(D, νk).

Proof. This is a direct consequence of (1.1.10) and (1.1.12).

Proposition 2.9.4. The functions wn(1−w)k∞n=0 form an orthonormal basis for L2(D, νk).The functions

φn =

(z − iz + i

)n(2i)k

(z + i)k| n = 0, 1, . . .

form an orthonormal basis for L2(H, µk).

Proof. Since the functions wn(1 − w)k on D and (2i)k(z − i)n/(z + i)n+k correspond toeach other via the Cayley transform, it is enough to prove that wn(1 − w)k∞n=0 form anorthonormal basis for L2(D, νk).

We first verify that ψn = wn(1 − w)k are square integrable on D with respect to themeasure νk. In polar coordinate w = reiθ, we have dudv = rdrdθ so that

νk =4(1− r2)k−2rdrdθ

|1− w|2k.

For every n, the integral

||ψn||2 ≤ 4

∫ 2π

0

∫ 1

0

r2n(1− r2)k−2drdθ

the latter integral being obviously convergent.We verify now that ψm and ψn are orthogonal with respect to the measure νk if m 6= n.

The integral ∫D

ψm(w)ψn(w)dw = 4

∫D

wmwn(1− r2)k−2drdθ

= 4

∫ 1

0

rm+n(1− r2)k−2

∫ 2π

0

ei(m−n)θdq

vanishes if m 6= n.

Proposition 2.9.5. The infinitesimal class of πk is the (g, K)-module D−k of highest weightk.

Proof. According to the classification of irreducible (g, K)-modules, it is enough to show thatthe K-type of πk is the set of integers −k − 2n|n ∈ N.

It is enough to show that kθ ∈ K acts on the vector

φn =

(z − iz + i

)n(2i)k

(z + i)k

by πk(k−1θ )φn = e2iπ(k+2n)θφn. This can be done by a direct calculation.

39

Let us do this calculation by writting down explicitly the action of G on L2(D, νk). Con-jugate by the Cayley matrix, we get

(2.9.2)1

2

[1 −i1 i

] [a bc d

] [1 1i −i

]=

[α ββ α

]where α = (a+ bi− ci+ d)/2 and β = (a− bi− ci− d)/2. We also have

cz + d =(ci− d)w + (ci+ d)

1− w

=(−α + β)w + (α− β)

1− w

If g =

[a bc d

], then we have

(2.9.3) πk(g−1)ψ(w) = ψ

(αw + β

βw + α

)(|α|2 − |β|2)k/2(1− w)k

((−α + β)w + (α− β))k

for every ψ ∈ L2(D, νk). If [α ββ α

]=

[eiθ 00 e−iθ

]then we have

πk(g−1)ψ(w) = ψ(e2iθw)

eikθ(1− w)k

(1− e2iθw)k.

For ψn = wn(1− w)k we have

πn(g−1)ψn(w) = ei(2n+k)θψn(w).

This implies that the K-type of πk is the set of integers −k − 2n|n ∈ N.

We attach to each holomorphic function f on H the smooth function φ on G

(2.9.4) φ(g) = f(gi)j(g, z)−k

with the automorphy factor

(2.9.5) j(g, z) = det(g)−1/2(cz + d).

if g =

[a bc d

].

We will make the formula for φ more explicit with help of the Iwasawa decomposition.Every matrix g ∈ G can be written uniquely in the form

(2.9.6) g =

[uy−1/2 0

0 uy−1/2

] [y x0 1

] [cos θ sin θ− sin θ cos θ

]where x, y, u ∈ R with y, u > 0 and θ ∈ R/2πZ. Apply g to i ∈ H, we get gi = x+ iy. Theautomorphy factor can be calculated to be j(g, i) = y−1/2e−iθ so that we get

(2.9.7) φ(g) = yk/2eikθf(x+ iy).

The above construction f 7→ φ defines a bijection between smooth functions f on H andsmooth functions φ on G that satisfy the equation

(2.9.8) φ(gkθ) = eikθφ(g)40

The opposite mapping φ 7→ f associates to a smooth function φ on G the function on H

(2.9.9) f(z) = y−k/2φ

(y−1/2

[y x0 1

])for every z = x+ iy with y > 0.

Proposition 2.9.6. The above construction f 7→ φ defines a bijection between holomorphicfunctions f on H and smooth functions φ on G that satisfy

(2.9.10) φ(gkθ) = eikθφ(g)

and

(2.9.11) LLφ = 0

where LL is the left differential derivation attached to the nilpotent matrix L of (2.5.1).

Proof. We only to prove that f is holomorphic if and only if φ is annihilated by LL. If f isholomorphic, it is annihilated by the operator

(2.9.12)∂

∂z=

1

2

(∂

∂x+

∂

∂y

).

According to 2.7.5, we have

LL = e−2iθ

(−iy ∂

∂x+ i

∂

∂y+i

2

∂

∂θ

)In order to prove that LL annihilates φ(g) = yk/2eikθf(x+ iy), it is enough to verify that itannihilates yk/2eikθ and f(x+ iy). One can check(

−iy ∂∂x

+ y∂

∂y+i

2

∂

∂θ

)(yk/2eikθ) = 0

by direct calculation. The factor f(x+iy) is also annihilated because of the Cauchy-Riemannequation.

Proposition 2.9.7. The map f 7→ φ defined in (2.9.4) defines an intertwining operatorfrom L2(H, µk) into L2(G). In particular, L2(H, µk) is a square integrable representation.

3. Automoprhic forms on SL2(R)

In this chapter, we denote G = SL2(R) and H the upper half plane. The main referenceis [1].

3.1. Siegel domains. Let Γ be a Fuschian group of first kind. Let us suppose that ∞ is acusp for Γ. The group B of triangular matrices is the stabilizer of ∞ for the homographicaction of G on P1(C). We have B = NA where N is the group of unipotent triangularmatrices and A the group of diagonal matrices. The action of A on the Lie algebra of Ndefines a character α : A→ R×. For each compact subset ΩN of N and t > 0, we define theSiegel domain

S = ωAtK

where At = y ∈ A | α(y) > t. If ω = |x| ≤ h, then the image of S is the upper halfband

x+ iy | |x| ≤ h and y > t.41

Assume that ω contains an interval in N = R which is a fundamental domain for the discretesubgroup Γ∞. Then Γ∞S is an upper half plane x+ iy | y > t.

We define by conjugation the notion of Siegel sets for other cusps.

Proposition 3.1.1. Let Γ be a Fuschian group of first kind and let u1, . . . , um be a set ofrepresentatives of Γ-equivalence of cusps. There exists a compact subset C of H and a Siegeldomain Si for each cusp such that C ∪

⋃mi=1 Si contain a fundamental domain of Γ.

Proof. For each cusp xi, we choose a Siegel domain Si whose image in Γ\H∗ is a neighbor-hood of xi. Since Γ\H∗ is compact by assumption, the complement of the union of imagesof Si is relatively compact. We can choose a compact C ⊂ H whose image contain it.

3.2. Growth condition. Assume that ∞ is a cusp for Γ and let Γ∞ is the stabilizer of ∞in Γ. Let φ be a function on G that is Γ∞-invariant on the left and has K-type k for theaction of K on the right i.e.

φ(γgkθ) = f(g)eikθ for all γ ∈ Γ∞, kθ ∈ K.

Then φ is said to be of moderate growth at ∞ if there exists λ > 0 so that

|φ(g)| < yλ

if g(i) = x + iy with x, y ∈ R and y > t for some fixed t. In other words, this inequalityis valid on a Siegel domain. The function φ is said to be of rapid decay if this inequality isvalid for all λ.

Definition 3.2.1. A function φ : Γ\G → C is said to be of moderate growth if it is ofmoderate growth at every cusps of Γ.

Lemma 3.2.2. If f has moderate growth of exponent λ then so does f ∗α for all α ∈ C∞c (G).

Lemma 3.2.3. Suppose that φ has moderate growth of exponent α on a Siegel domain S.Assume that φ1 = φ ∗ α for some α ∈ C∞c (G) then |Dφ(g)| < yα for any D ∈ U(g) andg ∈ S. We will say that φ1 as above has uniform moderate growth.

Proof. Since

Dφ1(g) = D(φ ∗ α)(g) = (φ ∗Dα)(g).

this lemma follows from the previous one.

There is another way to define the condition of moderate growth. Let

||g||2 = a2 + b2 + c2 + d2.

Proposition 3.2.4. A function φ : Γ\G has moderate at the cusp of and only if there existsm > 0 such that

|φ(g)| ≤ ||g||m.

The function φ is of rapid decay if this inequality is valid for all m.42

3.3. From modular forms to automorphic forms. Let f ∈ Mk(Γ) be a modular formof weight k with respect to Γ. The formula (2.9.4) defines a function φ on G that satisfiesthe equation and is annihilated by LL. Since the map f 7→ φ is G-equivariant, and f isΓ-invariant with respect to the action (1.6.1), φ is also invariant Γ-invariant. Thus φ definesa function Γ\G→ C that satisfies (2.9.4) and is annihilated by LL.

Proposition 3.3.1. Let f be a modular form of weight k and φ the smooth function on Gdefined by (2.9.4). Then φ satisfies the differential equation Ωφ = k(k − 2)φ.

Proof. It is clear for (2.7.16) that LHφ = kφ. In combining with LE−φ = 0, the calculation

Ωφ = (H2 + 2E+E− + 2E−E+)φ(3.3.1)

= k2φ+ 2[E−, E+]φ(3.3.2)

= k2φ− 2Hφ(3.3.3)

= k(k − 2)φ.(3.3.4)

proves our proposition.

Modular forms satisfy a condition of holomorphicity at the cusp, the associated functionφ will satisfy a growth condition at the cusps that we are about to describe.

Proposition 3.3.2. Let f be a modular form of weight k with respect to Γ. Then the functionφ on G defined as in (2.9.4) has moderate growth. If f is a cusp form, then φ has rapiddecay.

Proof. The function φ is given by the formula (2.9.7) phi(g) = yk/2eikθf(x + iy) where fadmits the Fourier expansion

f(x+ iy) =∞∑n=0

ane2niπxe−2nπy.

As y →∞, φ(g) has moderate growth. If a0 = 0, φ(g) has rapid decay as y →∞.

Definition 3.3.3. A smooth function φ : G→ C is an automorphic function for a Fuschiangroup of first kind Γ if

(1) φ has a unitary central character(2) φ(γg) = φ(g) for all γ ∈ G and g ∈ G,(3) φ is K-finite on the right,(4) φ is Z(U(g))-finite,(5) φ has moderate growth at the cusps,

Proposition 3.3.4. Automorphic form are real analytic.

Proof. Because φ is both Z(U(g))-finite and K-finite, it satisfies an elliptic differential equa-tion. It follows that φ is a real analytic function.

Proposition 3.3.5 (Harish-Chandra). Let φ be an automorphic form. For every neighbor-hood U of identity there exists α ∈ C∞c (G) with support contained in U so that φ = φ ∗ α.In particular, φ has uniform moderate growth.

43

Proof. Let V be the smallest closed G-invariant subspace containing φ. Since φ is K-finiteand Z(U(g))-finite, V is admissible i.e. for all all integer `, the eigenspace space V (`) forthe `-th power character of K is finite dimensional.

Let I∞c (G) denote the space of smooth function with compact support that are invariantwith respect to the conjugation by K. We will prove that there exists a α ∈ I∞c (G) witharbitrarily small support such that φ = φ ∗ α.

By assumption φ is a finite sum∑

`∈L φ` with φ` ∈ V (`) with L a finite subset of Z. Theconvolution φ 7→ φ∗α preserves L and defines a continuous linear map I∞c (U)→ End(L). Itsimage is vector subspace is a finite-dimensional vector space hence closed. Let αn ∈ I∞c (U)be a delta sequence whose image in End(L) tends to identity. It follows that identity can berepresented in the form φ 7→ φ ∗ α for some α ∈ I∞c (U).

3.4. L2-automorphic forms. The Peterson inner product (1.6.6) for cusp forms has anatural transposition to the framework work of automorphic forms. Let f, f ′ ∈ Sk(Γ) becusp forms of weight k with respect to a Fuschian group of first kind Γ. Let φ and φ′ befunctions on G associated with f and f ′ as (2.9.7). Then∫

Γ\Hf(z)f ′(z)yk

dxdy

y2=

∫ZΓ\G

φ(g)φ′(g)dg.

where dg is the Haar measure of G.

Definition 3.4.1. Let ω be a unitary character of Z. A L2-automorphic form with centralcharacter ω is a function φ : Γ\G→ C which transforms as ω under the action of the centerso that |φ(z)| is a square integrable function on ZΓ\G.

Proposition 3.4.2. Suppose that ∞ is a cusp of Γ. Let α ∈ Cc(G). Then there exists aconstant c so that

|(φ ∗ α)(g)| ≤ cy||φ||2for all f ∈ L2(Γ\G), all g ∈ NAtK and g(i) = x+ iy.

Proof. See [1, 5.7]. Assume that α is supported by C−1 where C is a compact subset of G.By definition

(3.4.1) (φ ∗ α)(g) =

∫G

φ(gx−1)α(x)dx

from which we derive the estimate

(3.4.2) |(φ ∗ α)(g)| ≤ ||α||∞∫gC

|φ(h)|dh.

Let CN and CA be compact subsets of N and A such that KC ⊂ CNCAK. After writing gin Iwasawa’s form

(3.4.3) g = y−1/2

[1 x0 1

] [y 00 1

] [cos θ sin θ− sin θ cos θ

]we have gC ⊂ gKC

(3.4.4) y−1/2

[1 x0 1

](Ad

[y 00 1

]CN)(

[y 00 1

]CA)K.

44

Assume that CN ∈ R and CA ∈ R+ are closed finite intervals. The image of this compactin H is a rectangle. The question is how many fundamental domains we need to cover thisrectangle. Since y > t for some fixed positive t the fundamental domain we need to ”coverin vertical direction” is bounded. The adjoint action of y on CN dilates in length with ratey. Therefore the number of fundamental domain we need to cover the rectangle is O(y). Weget the estimate

(3.4.5)

∫gC

|φ(h)|dh ≤ C1y||φ||1

for some constant C1. Using Cauchy-Schwarz inequality and the finiteness of the volume ofΓ\G, we get

(3.4.6)

∫gC

|φ(h)|dh ≤ C2y||φ||2.

THe proposition derives from this and (3.4.2).

Corollary 3.4.3. If φ ∈ L2(Γ\G) satisfying the condition (1, 2, 3, 4) of 3.3.3, then it alsosatisfies the moderate growth condition (5).

Proof. By 3.3.4 and 3.3.5, if φ satisfies (1, 2, 3, 4) of 3.3.3, then φ is real analytic end thereexists φ ∈ C∞c (G) with support arbitrarily close identity such that φ = φ ∗ α. We have theestimate

|φ(g)| ≤ cy||φ||2on the Siegel domain ΩNAtK of the cusp∞. We have also similar estimates for other cusps.It remains now to apply 3.1.1.

3.5. Constant terms. Suppose that ∞ is a cusp of Γ. Let B = NA is the subgroup oftriangular matrices in G, N its unipotent radical and ΓN = Γ ∩N . Let φ be a ΓN -invariantfunction on G that is locally integrable. The constant term φB of f is, by definition thefunction

(3.5.1) φB(g) =

∫ΓN\N

φ(ng)dn.

where the invariant measure is normalized so that the quotient ΓN\N has volume one.Since the constant term is defined by integration on the left, this operation commutes with

left-invariant differential operators as well as the convolution on the right

D(φB) = (Dφ)B and (φ ∗ α)B = φB ∗ αfor all D ∈ U(g) and α ∈ Cc(G).

Lemma 3.5.1. Let f be a modular form of weigh k and f =∑∞

n=0 anqn its Fourier expansion

at ∞. Let φ denote the associate function φ(g) = ykeikθf(x + iy). Then the constant termof φ is given by the formula φB(g) = ykeikθa0.

Proposition 3.5.2. Let L1cusp(Γ\G) be the subspace of L1(Γ\G) of integrable functions that

have vanishing constant terms at every cusp. Then L1cusp(Γ\G) is a closed subspace, stable

under the convolution on the right of Cc(G).

Proof. 45

Since Γ\G has finite volume, by applying the Cauchy-Schwarz inequality∫Γ\G|φ(x)|dx ≤

(∫Γ\G|φ(x)|2dx

)1/2(∫Γ\G

dx

)1/2

we have an inclusion L2(Γ\G) ⊂ L1(Γ\G), and the injection is continuous. We define

L2cusp(Γ\G) = L2(Γ\G) ∩ L1

cusp(Γ\G).

It follows that

Proposition 3.5.3. L2cusp(Γ\G) is a closed subspace of L2(Γ\G), stable under the convolu-

tion on the right of Cc(G).

Proposition 3.5.4. Let X1, X2, X3 be a basis of the Lie algebra of g. Let φ ∈ C1(ΓN\G).Then there exists a constant c > 0, independent of f , such that

|(φ− φB)(g)| ≤ cy−1

(3∑i=1

|LXiφ|B(g)

)where g(i) = x+ iy.

Proof. We have denoted

(3.5.2) R+ =

[0 10 0

]which is a generator of the Lie algebra of N . We have

(3.5.3) (φB − φ)(g) =

∫ 1

0

(φ(etR+g)− φ(g))dt.

But clearly,

(3.5.4) φ(etR+g)− φ(g) =

∫ t

0

(RR+φ)(euR+g)du

where RR+ is the right invariant derivation attached to R+. We want to convert the aboveexpression to left invariant derivation.

(RR+φ)(euR+g) =d

dtφ(euR+etR+g)|t=0 =

d

dtφ(euR+getAdg−1R+)|t=0

We will use again the Iwasawa decomposition

g = y−1/2

[1 x0 1

] [y 00 1

] [cos θ sin θ− sin θ cos θ

]we have

euR+getAdg−1R+ = euR+gety−1Adk−1

θ R+ .

There are smooth functions c1, c2, c3 on K so that

k−1θ R+ =

3∑i=1

ci(kθ)Xi

46

We have

(RR+φ)(euR+g) = y−1

3∑i=1

ci(kθ)(LXiφ)(euR+g).

The continuous functions on the compact set K are bounded by some constant c so that

(3.5.5) |(R−R+φ)(euR+g)| ≤ cy−1

3∑i=1

|(LXiφ)(euR+g)|.

It follows that

|∫ t

0

(RR+φ)(euR+g)du| ≤ cy−1

3∑i=1

∫ 1

0

|(LXiφ)(euR+g)|du

= cy−1

3∑i=1

|LXiφ|B(g)

from which derives our desired estimate.

3.6. Convolution on the cuspidal spectrum.

Proposition 3.6.1. Let α ∈ C1c (G). There exists a constant c(α) so that

|(φ ∗ α)(g)| ≤ c(α)||φ||2

for all φ ∈ L2cusp(Γ\G).

Proof. Let P be a cuspidal subgroup of Γ and S s Siegel set relative to P . By 3.4.2, thereexists a constant c1(α), depending only on α such that

(3.6.1) |(φ ∗ α)(g)| ≤ c1(α)y||φ||2

for g ∈ S and g(i) = x+ iy. If X ∈ g and LX is the associated left invariant derivation, wehave LX(φ ∗ α) = φ ∗ LX(α). Thus there exists a constant c(α,X) such that

(3.6.2) |(LX(φ ∗ α))(g)| ≤ c(α,X)y||φ||2.

for g ∈ S. Since the translation on the left by the unipotent radical N of B keeps thecoordinate y of g invariant, the integration over N gives

(3.6.3) |(LX(φ ∗ α))B(g)| ≤ c(α,X)y||φ||2.

We are now applying the inequality to the elements X1, X2, X3 of a basis of g. By as-sumption φB = 0 so that (φ ∗ α)B = 0, we have the estimate

(3.6.4) |(φ ∗ α)(g)| ≤ c(α)||φ||2

for some constant c(α) that depends only on α.

Proposition 3.6.2. For all α ∈ C2c (G), the convolution φ 7→ φ∗α defines a compact operator

on L2cusp(Γ\G).

47

Proof. Let φn be a sequence of function on Γ\G such that ||φn||2 ≤ 1. The estimate 3.6.1implies that there exists a contant c(α) so that |(φn ∗α)(g)| < c(α) for all g and all n. Thusthe sequence φn ∗α is bounded with respect to the uniform norm. The same estimate appliesto LX(φn∗α) for all left invariant derivation LX . It implies that this family is equicontinuous.Now by application of the Arzela-Ascoli theorem and we can extract a subsequence φn′ ∗ αthat converges locally uniformly to a function φ′ that is continuous and bounded. SinceΓ\G has finite volume, φ′ is square integrable. It remains only to recall that L2

cusp(Γ\G) is

a closed subspace of L2(G) so that φ′ ∈ L2cusp(Γ\G).

The following general statement is due to Gelfand, Graev and PS. The following proof isdue to Langlands. See [11].

Proposition 3.6.3. Let (π,H) be a unitary representation of G. If there exists a deltasequence φn on G such that π(φn) is compact operator on H then H is a Hilbert direct sumof irreducible unitary representations which appear with finite multiplicity.

Proof. The idea is to use the spectral theory of compact self-adjoint operators. Recall that acompact self-adjoint operators have eigenvalues, the set of eigenvalues has no accumulationpoint except 0, and for each non zero eigenvalue, the corresponding eigenspace is finitedimensional.

Let (π1,H1) be an irreducible unitary representation of G. Since φn is a delta sequence,π1(φn) 6= 0 for some n. There exists λ 6= 0 so that the λ-eigenspace of π1(φn) is non-zero. Since the λ-eigenspace of the compact self-adjoint operator π(φn) is finite dimensional,(π1,H1) appears in (π,H) with finite multiplicity.

We will now prove that H has at least one irreducible subspace with the help of the compactself-adoint operators π(φn). For n large enough π(φn) 6= 0 has a nonzero eigenvalue λ. Let Vdenote the λ-eigenspace of π(φn) which is finite dimensional. Consider the non zero subspaceV ′ ⊂ V so that V ′ = V ∩ H′ where H′ is a closed invariant subspace of H. Since V is finitedimensional, this family has minimal element for the inclusion order. Let V1 is a minimalelement of this family.

Let H1 denote the intersection of all closed invariant subspaces H′ ⊂ H such that H′ ∩V =V1. We will prove that H1 is irreducible. If it is not, we have a direct decompositionH1 = A ⊕ B where A,B are closed invariant subspaces. Since A and B are stable underπ(φn), if we decompose a vector v ∈ V1 as v = a⊕ b with a ∈ A and b ∈ B, then a, b are alsoλ-eigenvectors. It follows that

V1 = (V1 ∩ A)⊕ (V1 ∩B).

The minimality of V1 implies that either V1 ∩ A = V1 or V1 ∩ B = V1. But this contradictswith the minimality of H1.

Let H′ be the closure of the sum of all irreducible subspaces in H and let H′′ be theorthogonal complement of V . If H′′ 6= 0, it contains an irreducible subspace. This wouldcontradict with the contradict with the definition of H′. Therefore H′′ = 0 and H = H′.

Theorem 3.6.4. The Hilbert space L2cusp(Γ\G) decomposes as a direct sum of irreducible

unitary representations of G, each occurs with finite multiplicity.

3.7. Duality theorem.

48

Proposition 3.7.1 (Gelfand, Graev, PS). There is an isomorphism

Sk(Γ) = HomG(D+k , L

2cusp(Γ\G)).

Proof. Let fix a lowest weight vector vk of D+k . It is of K-weight k and annihilated by LL.

Let α : D+k → L2

cusp(Γ\G). Then α(vk) is an analytic L2-function that is of K-weight K,annihilated by LL. By 2.9.6, φ comes from a cuspidal modular form f of weight k.

Let s be a purely imaginary number and let Ps be the unitary principal series. Let v0 bea vector of weight 0 in Ps. For any intertwining operator

α ∈ HomG(Ps, L2cusp(Γ\G))

the function α(v0) is a cuspidal analytic function on Γ\G that is right K-invariant. It satisfiesthe differential equation Ωφ = (s2− 1)φ. It defines an analytic function f : Γ\H→ C whichis an eigenvalue of the Laplacian

(3.7.1) ∆ = −y2

(∂2

∂x2+

∂2

∂y2

)then ∆f =

1− s2

4f.

Those are Maass modular forms, eigenvector for the Laplacians with eigenvalues λ ≥ 1/4.

Conjecture 3.7.2 (Selberg). For all s ∈ [−1, 1], and Cs the complementary series, we have

HomG(Cs, L2cusp(Γ\G)) = 0

for all congruence subgroup.

This is actually false for some non congruence subgroup.

4. Smooth representations of GL2(Qp)

In this chapter, G = GL2(F ) where F is a local nonarchimedean field. Let OF denote thering of integers of F .

4.1. Cartan and Iwasawa decompositions. As in the case of real Lie groups, Cartanand Iwasawa decomposition plays an important role in their representation theory. In thecase of GL2, these decompositions can be described with linear algebras over the p-adicfields. Let B = AN denote the subgroup of upper triangular matrices, A is the groupof diagonal matrices and N the group of unipotent upper triangular matrices. The groupK(1) = GL2(OF ) is a maximal compact open subgroup of G. Every compact subgroupsubgroup of G is conjugate to a subgroup of K(1) of finite index.

Proposition 4.1.1 (Cartan decomposition).

G =⊔

d1≤d2,di∈Z

K(1)

[pd1 00 pd2

]K(1).

Proof. Let X denote the set of all lattices L ⊂ F 2. The group G acts transitively on Xand its stabilizer at the standard lattice L0 = O2

F is K. It follows that X = G/K(1). TheCartan decomposition now follows from the theory of elementary divisors : for every latticeL ∈ X, there exist a basis x1, x2 of L0 so that L = pd1x1OF ⊕ pd2x2OF where d1, d2 areintegers satisfying d1 ≤ d2.

49

Proposition 4.1.2 (Iwasawa decomposition).

G = BK(1) =⊔

d1,d2∈Z

N

[pd1 00 pd2

]K(1).

Proof. Let x1, x2 denote the standard basis of F . The group B upper triangular matrices isthe stabilizer of the line generated by x1. For every lattice L ∈ X, we can define two integersd1, d2 ∈ Z as follows : L1 = L ∩ x1F being a lattice of x1F , must be of the form pd1x1OFfor some integer d1; the quotient L/L1 being naturally a lattice in x2F , must be of the formpd2x2OF . It is not hard to check that there exists a triangular matrix n ∈ N so that

L = n

[pd1 00 pd2

]L0.

The Iwasawa decomposition follows.

4.2. Hilbert representations. Let C∞c (G) denote the space of locally constant functionwith compact support. With a choice of Haar measure on G, C∞c (G) is an algebra withrespect to the convolution product. Be aware that this algebra is not equipped wit ha unit.We observe that every function f ∈ C∞c (G) is biinvariant with respect to some compactopen subgroup K. In other words, if HK denotes the algebra, with unit, of K-biinvariantfunctions with compact support, then

C∞c (G) =⋃K

HK

the union being taken over all compact open subgroups of G.Let (π,H) be a Hilbert representation of G. The algebra C∞c (G) acts on H by

π(φ)v =

∫G

φ(g)π(g)vdg.

If φ ∈ HK , then π(φ)v ∈ HK .A vector v ∈ H is said to be smooth if is stabilized by a compact open subgroup of G. Let

V be the subspace of smooth vectors in H. The group G acts continuously on V equippedwith the discrete topology. For all compact open subgroup K of G, the subspace V K ofK-invariant vectors is a module of HK .

Proposition 4.2.1. Let (π,H) be a Hilbert representation of G and let V denote its subspaceof smooth vectors. Then H is irreducible if and only if V an irreducible smooth representationof G, if and only if for every compact open subgroup K of G, V K is an irreducible HK-module.

Smooth representations of p-adic groups play a similar roles to (g, K)-modules of real Liegroups.

4.3. Unramified representations.

Definition 4.3.1. An irreducible smooth representation of G is said to be unramified ifV K(1) 6= 0 where K(1) is the maximal compact subgroup of G.

Unramified representations are just irreducible H-modules. We have a complete descrip-tion of the algebra H that allows us to describe its irreducible modules. By using Gelfand’strick, one sees that H is commutative. In fact, we have a much more precise description ofthis algebra using Harish-Chandra’s constant terms.

50

Theorem 4.3.2 (Satake isomorphism). For every φ ∈ H, the constant term of φ along Bis the function φB : A→ C given by the formula

φB(a) = δB(a)1/2

∫N

φ(an)dn

the Haar measure dn on N is choosen so that K ∩N has measure one. The constant termmap defines an isomorphism from H onto the algebra of compactly supporte functions on Ainvariant under A ∩K and under the action of the Weyl group W .

Proof. Orbital integral of a diagonal elements can be expressed as constant terms.

Corollary 4.3.3. Let A denote the complex dual torus of A. There is an isomorphism ofalgebras between H and C[A]W . In particular, the unramified representations of G are in

natural bijection with A/W .

Corollary 4.3.4. Let G = GL2(C) denote the Langlands dual group of G = GL2(F ). The

unramified representations of G are classified by the semisimple conjugacy classes of G.

The trivial representation of G is certainly unramified. It corresponds to the homomor-phism H → C given by

φ 7→∫G

φ(g)dg.

The associated conjugacy semisimple class in GL2(C) is the class of the matrix

(4.3.1)

[p 00 1

]Proposition 4.3.5. If V is an unramified representation then its contragredient V ′ is alsoan unramified.

Proof. The linear form v 7→∫K(1)

π(k)v is K(1)-invariant.

Proposition 4.3.6. Let V be an unramified representation of V and v ∈ V be a K(1)-invariant vector.

4.4. Smooth admissible representations. A representation of G on a complex vectorspace V is called smooth if every vector v ∈ V is stabilized by a compact open subgroup K.

Lemma 4.4.1. Let V ′ be a subspace of V that stable under the action of K and irreducibleas representation of K. Then V ′ is finite-dimensional.

Proof. A vector v′ ∈ V ′ is stabilized by a finite index subgroup K ′ of K. The subspace ofV ′ generated by gv for a set of representatives K/K ′ is K-stable. Since V ′ is irreducible, V ′

is generated by those vectors and hence finite-dimensional.

Let K be a compact open subgroup of G. Let V be a smooth representation of G. Forevery finite dimensional irreducible representation (ρ, Vρ) of K be the sum of all K-invariantsubspaces of V which are isomorphic to Vρ. The V can be decomposed as an algebraic directsum

V =⊕ρ∈K

V (ρ).

51

V is said to admissible if and only if for every ρ ∈ K, V (ρ) is finite dimensional. A smoothrepresentation of G is admissible if and only if dim(V K) is finite for all compact open sub-groups K of G.

Theorem 4.4.2 (Harish-Chandra). All irreducible smooth representations of G are admis-sible.

Let V be a smooth representation of G. The contragredient representation V ′ is the spaceof smooth vectors in the vector space of linear forms on V .

Proposition 4.4.3. Let V be a smooth admissible representation of G. Let V =⊕

ρ V (ρ)denote the decomposition of V into isotypical components with respect to a compact opensubgroup K of G. Then the contragredient of G is an algebraic direct sum

V ′ =⊕

V (ρ)′

where V (ρ)′ = Hom(V (ρ),C). In particular, V ′ is admissible if V is. In that case, we haveV = (V ′)′.

Theorem 4.4.4 (Gelfand-Kazhdan). The group G = GL2(Qp) is equipped with the outerautomorphism

g 7→ >g−1.

Let (π, V ) be a irreducible admissible representation of G. Then the contragredient (π′, V ′)is isomorphic with (π′′, V ) where π′′(g) = π( >g−1).

This theorem is surprisingly difficult to prove. We will need in particular to introduce thenotion of character of an admissible representation. For every φ ∈ C∞c (G), there exists opencompact subgroup G so that φ ∈ HK . The operator π(φ) : V → V has image contained inthe finite-dimensional space HK . The number

Trπ(φ) = Tr(π(φ)|V K )

does not depend on the choice of K. We thus define a linear map

Trπ : C∞c (G)→ Chence a distribution on G.

Proposition 4.4.5. Two irreducible admissible representations (π1, V1) and (π2, V2) havingthe same character Trπ1 = Trπ2, are isomorphic.

Proof. Let K be a compact open subgroup of G small enough so that V1 and V2 have nonzeroK-invariant vectors. It follows that V K

1 and V K2 are irreducibleHK-modules having the same

function trace. It follows that they are isomorphic as HK-modules (Bourbaki, see Lang’sdiscussion on the Jacobson density theorem). Hence V1 and V2 are isomorphic as smoothrepresentations of G.

In order to prove the Gelfand-Kazhdan theorem, we need to prove that (π′, V ′) and (π′′, V )have the same character. Let φ ∈ C∞c (G). Let denote φ′(g) = φ(g−1) and φ′′(g) = φ( >g−1).By definition, π′(φ) and π(φ′) are adjoint operators so that they have the same trace. Alsoby definition, we have π′′(φ) = π(φ′′). It is thus enough to prove that

Trπ(φ′) = Trπ(φ′′)

or in other words that the distribution Trπ is transposition invariant.52

Proposition 4.4.6. Over GL(2, F ), a conjugation invariant distribution is also transpositioninvariant.

4.5. Analysis on totally disconnected spaces. A td-space is a separated topologicalspace in which every point has a basis of neighborhood given by compact open subsets.

Lemma 4.5.1. Let X be a td-space. Let Y be a compact subset of X. From every coveringof X by a open subsets of X, we can extract a finite family that, after refinement, forms acovering of Y by disjoint open subsets of X.

We will denote S(X) = C∞c (X) the space of complex valued locally constant functionswith compact support on X. For every compact open subset U of X, we denote 1U thecharacteristic function of U . All function φ ∈ S(X) is a finite linear combination of suchfunctions 1U .

The vector space S(X) has a structure of algebra with respect to the pointwise multipli-cation. For every x ∈ X, let mx the ideal of S(X) of functions vanishing at x. Every elementφ ∈ mx is a finite linear combination of 1U where U is a compact open not containing x. Infact, we have mx = S(X − x). We have an exact sequence of C-vector spaces

0→ mx → S(X)→ C→ 0.

In fact, all morphism in this sequence are ring morphism. We have more generally thefollowing proposition.

Lemma 4.5.2. Let Y be a closed subset of X and U = X − Y . The we have an exactsequence

0→ S(U)→ S(X)→ S(Y )→ 0.

Proof. Only the surjectivity S(X) → S(Y ) is not totally obvious. It follows in fact from4.5.1.

Definition 4.5.3. A S(X)-module M is called smooth if for every m ∈ M , there exists acompact open subset U of X such that 1Um = m.

Let M be a smooth S(X)-module. We can define the fiber of M at a point x as follows

Mx = M/mxM

where mxM is the submodule of M generated by the elements φm with φ ∈ mx and m ∈M .

Lemma 4.5.4. Let M be a smooth S(X)-module and x ∈ X. An element m ∈ mxM if andonly if for all small enough compact open U containing x so that 1Um = 0.

Proof. If m = 1Vm′ where V is a compact open subset not containing x, then for all compact

open U containing x but disjoint from V we have 1Um = 1U1Vm′ = 0. Let m ∈ M such

that 1Um = 0 for all small enough compact open U containing x. By smoothness hypothesisthere exists a compact open subset V such that 1Vm = m. If x /∈ V , we are done. If x ∈ V ,there exists a compact open U with x ∈ U ⊂ V such that 1Um = 0. Then 1V−Um = m.

Lemma 4.5.5. Let M be a smooth S(X)-module. Suppose that Mx = 0 for all x ∈ X. ThenM = 0.

53

Proof. Let m ∈ M be an arbitrary element. There exists a compact open subset V so that1Vm = m. For every x ∈ V , since Mx = 0 there exists a compact open subset Ux containingx and contained in V such that 1Uxx = 0. We can even require that 1U ′xm = 0 for all compactopen U ′x contained in Ux containing x. It follows that 1U ′x = 0 for all compact open subsetU ′x of Ux that may contain x or not because if x /∈ U ′x, we can write 1U ′x = 1Ux − 1Ux−U ′x . Byapplying 4.5.1, we can write 1V as finite sum of 1U ′x which implies m = 0.

Lemma 4.5.6. Let 0→M ′ →M →M ′′ → 0 be an exact sequence of smooth S(X)-modules.Then for every x ∈ X, we have an exact sequence

0→M ′x →Mx →M ′′

x → 0.

Proof. The right exactness is a general property of tensor product. Here, we also have theleft exactness because Mx is not only the residual fiber but the fiber itself. More concretely,let m′ ∈M ′ whose image m ∈M satisfies mx = 0. There exists a compact open U containingx such that 1Um = 0. This implies that 1Um

′ = 0 and of course m′x = 0.

There are plenty of natural smooth S(X)-module.

Proposition 4.5.7. Let Y → X be a continuous map of td-spaces. Then S(Y ) is naturallya smooth S(X)-module. Moreover, for every x ∈ X, we have S(Y )x = S(Yx).

Proof. Let φ be a locally constant with compact support U in Y . The image of U in X is acompact subset of X that can be covered by a compact open subset U of X. Then we have1Uφ = φ.

We have an exact sequence

0→ S(Y − Yx)→ S(Y )→ S(Yx)→ 0

where S(Y − Yx) = mxS(Y ). It follows that S(Yx) = S(Y )x.

Proposition 4.5.8. Let Y → X be a continuous map between td-spaces. Let G be a td-group acting on Y , preserving the map Y → X and acting transitively on the fibers Yx. LetS(Y )G be the quotient of S(Y ) by the subspace S(Y )(G) generated by the functions of theform gφ − φ. Then S(Y )G is a smooth S(X)-module whose fibers S(Y )G,x is of dimensionat most one.

Proof. Since the action of G on S(Y ) commute with multiplication by S(X), the subspaceS(Y )(G) is a S(X)-submodule. The module quotient is automatically smooth because S(Y )is a smooth S(X)-module.

The fiber S(Y )G,x is the quotient of S(Y ) by the subspace generated by mxS(Y ) andS(Y )(G). It is thus the quotient of S(Yx) by the image of S(Y )(G) in S(Yx) which isS(Yx)(G).

Now, since G acts simply transitively on S(Yx), the space of G-invariant distribution onYx is of dimension at most one.

Proof. (of 4.4.6) Let Y denote G − Z the space of non central 2 × 2-matrices. Let Y →X = F × F× denote the characteristic polynomial g 7→ (Tr(g), det(g)). The group G actstransitively on the fibers of Y → X by conjugation. For every y ∈ Y , the centralizerGy is commutative, in particular unimodular, hence the quotient G/Gy has a G-invariantmeasure. The S(X)-module S(Y )G has fibers of dimension one. The transposition defines aninvolution on this space that acts as identity on every fiber. It follows from 4.5.5 that it acts

54

trivially on the whole module S(Y )G. In other words, a conjugation invariant distributionon Y in transposition invariant. Since the similar statement is trivial for Z, the statementfollows from the exact sequence 4.5.2.

The same technique of proving 4.4.6 will permit us to prove an important property ofBessel distributions. Let us fix a non trivial additive charater ψ : F → C×. We call Besselfunctions the locally constant functions b : G→ C satisfying the transformation property

b(n1gn2) = ψ(n1)ψ(n2)b(g).

We will be interested in the more general notion of Bessel distributions b : S(G) → Csatisfying

b(ln−11rn2φ) = ψ(n1)ψ(n2)b(φ).

Recall that by definition ln−11rn2φ(g) = φ(n1gn2).

Let θ : G → G be the anti-involution defined by θ(g) = w0>gw0 where w0 is the permu-

tation matrix. For every n ∈ N we have θ(n) = n. The involution θ induces an involutionon S(G) and on the space of distribution.

Proposition 4.5.9. If b is a Bessel distribution, then we have θ(b) = b.

4.6. Uniqueness of the Whittaker model. Let N be the subgroup of unipotent uppertriangular matrices. Let ψ : F → C× be a non trivial character of F . A Whittaker functionwith respect to ψ is a locally constant function φ : G → C such that φ(ng) = ψ(n)φ(g) forall n ∈ N and g ∈ G and there exists a compact open subset C ⊂ G so that the support ofφ is contained in NC. The space of all Whittaker functions is

cIndGN(ψ)

the compact induction of ψ. This space affords with a representation of G by translation onthe right.

Let (π, V ) be an irreducible admissible representation of G. A Whittaker model of G is anintertwining operator α : V → cIndGN(ψ). For all such α, we have a linear form lα : V → Cgiven by lα(v) = α(v)(1G) which satisfies la(π(n)v) = ψ(n)la(v). In general, la is not fixed byany compact open subgroup so that lα does not belong to the contragredient representationV ′.

We have in fact a bijection

HomG(V, cIndGN(ψ)) = HomN(V, ψ)

by application of the Frobenius reciprocity. Let Vψ(N) be the subspace of V spanned by thevectors of the form π(n)v − ψ(n)v with n ∈ N and v ∈ V . Let VN,ψ = V/Vψ(N). Everylinear functional l ∈ HomN(V, ψ) must factorize though VN,ψ and in fact we have

HomN(V, ψ) = Hom(VN,ψ,C).

If these vector spaces are nonzero then we say that V admits a Whittaker model.

Lemma 4.6.1. If V admits a Whittaker model with for a non trivial additive characterψ : F → C×, then it has Whittaker model for any non trivial additive character.

Proof. The group A of diagonal matrices acts transitively on the set of non trivial charactersof N .

55

Proposition 4.6.2. Let V be an irreducible admissible representation. If V admits a modelof Whittaker then so does its contragredient V ′.

Proof. We know by Gelfand and Kazhdan that the contragredient representation (π′, V ′) isisomorphic to (p′′, V ) where π′′(g) = π( >g−1). If l : V → C is a Whittaker functional of πwith respect to (N,ψ) then it is a Whittaker functional for π′′ with respect to >(N,ψ)−1.After conjugation, we will get a Whittaker functionalof π′′ with respect to (N,ψ).

Theorem 4.6.3. An irreducible admissible representation admits at most one Whittakermodel. In other words

dimG(V, cIndGN(ψ)) ≤ 1.

The theorem follows from 4.6.2 and the following proposition.

Proposition 4.6.4. Let V be an irreducible admissible representation. Then we have

dimG(V, cIndGN(ψ)) dimG(V ′, cIndGN(ψ)) ≤ 1.

Proof. Let l′ : V → C and l : V ′ → C be nonzero Whittaker functionals of V and V ′. If l, l′

are smooth linear form i.e. l′ ∈ V ′ and l ∈ V we can define the matrix coefficient functionG→ C

c(g) = 〈l, π′(g)l′〉that satisfies

c(n1gn2) = ψ(n1)ψ(n2)c(g).

But in fact l, l′ are not smooth vectors, we will have to define c as a Bessel distribution.For every φ ∈ S(G), we consider the linear form π′(φ)l′ : V → C defined by v 7→ l′(π(φ′)v)

where φ′ is the function φ(g) = φ′(g−1). This is a smooth linear form on V so that π′(φ)l′ ∈V ′. In the same way, for every φ ∈ S(G), we have a smooth vector π(φ)l ∈ V . One cancheck that

(4.6.1) π(rnφ)l = ψ(n)π(φ)l and π′(rnφ)l′ = ψ(n′)π′(φ)l′

We also have

π(lgφ)l = π(g)π(φ)l and π′(lgφ)l′ = π′(g)π′(φ)l

so that the map φ 7→ π(φ)l and φ 7→ π′(φ)l′ are intertwining operators S(G) → V andS(G)→ V ′ respectively.

We have a bilinear form on S(G)

b(φ1, φ2) = 〈π(φ1)l, π′(φ2)l′〉

which satisfies

b(lgφ1, lgφ2) = b(φ1, φ2).

Note that S(G) ⊗ S(G) = S(G × G). Consider the diagonal action of G acting by lefttranslation on G × G given by the formula g(g1, g2) = (gg1, gg2). The quotient G\(G × G)can be identified with G by the map G×G→ G given by (g1, g2) 7→ g−1

1 g2. The integrationalong the fibers of this map defines a linear map

S(G)⊗ S(G)→ S(G)56

which is (φ1, φ2) 7→ φ′1∗φ2 where φ′1(g) = φ1(g−1). Because the linear form φ1⊗φ2 7→ b(φ1, φ2)is G-invariant with respect to the diagonal action of G by lest translation on G × G, thereexists a unique linear form

c : S(G)→ Csuch that c(φ′1 ∗ φ2) = b(φ1, φ2).

Because of (4.6.1), c is a Bessel distribution

c(ln1rn2φ) = ψ−1(n1)ψ(n2)c(φ).

By applying 4.5.9, we know that θ(c) = c where θ is the induced operaotr on the space ofdistribution of θ(g) = w0

>gw0. Thus

c(φ′1 ∗ φ2) = c(θ(φ′1 ∗ φ2)) = c(θ(φ2) ∗ θ(φ′1))

thus

b(φ1, φ2) = b(θ(φ′2), θ(φ′1)).

If φ2 is in the kernel of the intertwining operator l′ : S(G) → V ′ the θ(φ′2) is in the kernelof l : S(G) → V . It follows that l′ determines the kernel of l. By Schur lemma, as V isirreducible, l′ determines l up to a scalar. The proposition follows.

Lemma 4.6.5. For every compact subgroup N0 ⊂ N , we set

Vψ(N0) = v ∈ V |∫N0

ψ−1(n)π(n)vdn = 0

Then we have Vψ(N) =⋃N0Vψ(N0).

Proof. An element v ∈ Vψ(N) is a linear combination of vectors of the form π(n0)v0−ψ(n0)v0.Every n0 ∈ N , there exists a compact subgroup N0 of N such that n0 ∈ N0. Then we have

(4.6.2)

∫N0

ψ−1(n)π(n)π(n0)v0 − ψ(n0)v0dn = 0

thus π(n0)v0 − ψ(n0)v0 ∈ Vψ(N0). It follows that Vψ(N) ⊂⋃N0Vψ(N0).

Let v ∈ V (N0). There exists a compact open subgroup N1 ⊂ N0 that stabilizes v. Byshrinking N1 we can also assume that ψ is trivial on N1. Then we have

(4.6.3)

∫N0

ψ−1(n)π(n)vdn =∑

n∈N0/N1

ψ−1(n)π(n)v = 0

Let r denote the cardinal N0/N1. Then we have

v =1

r

∑n∈N0/N1

ψ−1(n)(ψ(n)v − π(n)v)

and thus v ∈ Vψ(N).

Let V be an admissible representation of G. The above lemma suggest that Whittakerfunctionals are fibers of V with respect to a structure of S(F )-module that we are going todefine. Choose a non trivial additive character ψ : F → C×. The Fourier transform

φ(x) =

∫F

φ(xy)ψ(x)dx

57

defines an isomorphism S(F )→ S(F ). There exists a unique choice of Haar measure dx so

thatˆφ(x) = φ(−x). The Fourier transform turns pointwise multiplication into convolution

product. It follows that if we let φ ∈ S(F ) act on V by

(4.6.4) ι(φ)v :=

∫N

φ(n)π(n)vdn

then V has a structure of S(F )-module.

Lemma 4.6.6. V is a smooth S(F )-module.

Proof. Since V is smooth as representation of G, there exists a compact open subgroup N0

that stabilizes v. The Fourier transform of 1N0 is of the form 1U for certain compact opensubgroup U . Then we have ι(1U)v = π(1N0)v = v.

Lemma 4.6.7. For every a ∈ F×, Va and VN,ψa, where ψa is the additive character x 7→ψ(ax) are the same quotient of V .

Proof. The description of Vψa(N) given in 4.6.5 identifies this submodule with V (a). Itfollows an identification with quotient modules Va = VN,ψa .

Proposition 4.6.8. Let 0 → M ′ → M → M ′′ → 0 be an exact sequences of admissiblerepresentations. Then we have an exact sequence

0→M ′N,ψ →MN,ψ →M ′′

N,ψ → 0.

Proposition 4.6.9. Let V be an irreducible admissible representation. Then G has Whit-taker model.

Proof. Assume that V does not have Whittaker model. Then Va = VN,ψa = 0. It followsthat V = V0. The fiber V0 is the Jacquet module of V that is finite dimensional as we willlater show.

4.7. Jacquet module. Let (π, V ) be a smooth representation of G. The space V (N) is thesubspace of V spanned by the vectors of the form π(n)v − v with n ∈ N and v ∈ V . For acompact subgroup N0 ⊂ N define V (N0) to be

V (N0) = v ∈ V |∫N0

π(n)vdn = 0.

As in 4.6.5, we have V (N) to be V (N) =⋃N0V (N0). Let define the Jacquet module

VN = V/V (N) which is also the fiber V0 of V as S(G)-module defined in (4.6.4).The Jacquet module VN is equipped with an action πN of A = B/N . Observe that if a

vector v ∈ V is fixed by a compact open subgroup K of G, its image v in VN is necessarilyfixed by K ∩A. Hence, (πN , VN) is a smooth representation of A provided that (π, V ) to bea smooth representation of G.

Proposition 4.7.1. VN is an admissible representation of A.

The proof of the theorem is based on the notion of Iwahori decomposition. This decom-position is valid for an important family of compact open subgroups of G. Let denote

58

K(pr) =

[a bc d

]∼=[1 00 1

]mod pn

K1(pr) =

[a bc d

]∼=[1 ∗0 1

]mod pn

K0(pr) =

[a bc d

]∼=[∗ ∗0 ∗

]mod pn

For r ≥ 1, all these groups satisfy the following properties

Lemma 4.7.2. Let K be one of the above compact open subgroups for an integer r ≥ 1.Then every k ∈ K can be written uniquely in the form r = nan− where n ∈ NK = N ∩K,a ∈ AK = A ∩K and n− ∈ N−K = N− ∩K is a unipotent lower triangular matrix.

Proof. The existence of the Iwahori decomposition in the above example is d ∈ O×F in allcases.

Proposition 4.7.3. Let K be a compact open subgroup that admit an Iwahori decompositionas above. Then the canonical map

V K → V AKN

is surjective.

Proof. Let E be a finite dimensional subspace of V AKN . Since AK is a compact subset,

there exists a finite dimensional subspace E ′ of V AK that maps onto E. Since E ′ is finitedimensinoal there exists a compact subgroup N−0 of N− that fixes N−. There exists a ∈ Asuch that π(a)N0π

−1(a) ⊂ NK hence AKN−K fixes π(a)E ′. In other words, for every v ∈

π(a)E ′ we have ∫AK

∫NK

π(a)π(n−)vdn−da = v.

It follows that ∫NK

∫NK

π(n)π(a)π(n−)vdn−dadn =

∫NK

π(n)vdn.

The application v 7→ π(1NK )v send π(a)E ′ into V K . It follows that the image of V K in VNcontains πN(a)E. But since πN(a) is invertible in VN , it follows that

dim(E) = dim(πN(a)E) ≤ dim(V K).

Since E have been chosen as an arbitrary finite dimensional subspace of V AKN , it follows that

V AKN is finite and bounded by dim(V K). We can thus take E = V AK

N hence V K → V AKN is

surjective.

In fact, the structure of the map V K 7→ V AKN can be made very precise because it is linear

over a rather large algebra. Let A− denote the sub semigroup of A of diagonal matrices(a1, a2) such that a1a

−12 ∈ OF in other words A− = a ∈ A | |α(α)| ≤ 1 where α is the

positive root. For a ∈ A, the conjugation by A contracts N and dilates N−. More precisely,we have

(4.7.1) aNKa−1 ⊂ NK and a−1N−Ka ⊂ N−K

Let C[A−/AK ] be the algebra generated by the monoid A−/AK .59

Lemma 4.7.4. Let K be as above. The application a 7→ 1KaK defines a ring morphismC[A−/AK ]→ HK.

Proof. Using (4.7.1), we havea1Ka2 ⊂ Ka1a2K

for all a1, a2 ∈ A−. The lemma follows.

Proposition 4.7.5. The map V K → V AKN is C[A−/AK ]-linear. There is a decomposition

V K = E0⊕E1 where π(a) = 0 on E0 and is bijective on E1. The map V K → V AKN is null on

E0 and induces a bijection E1 → V AKN .

Corollary 4.7.6. The map V I → V AIN is a bijection if I = K0(p) is the Iwahori subgroup.

Proof. In this case 1IaI is an invertible element of the Hecke-Iwahori algebra HI .

In our case G = GL2(F ), a much stronger statement is true.

Proposition 4.7.7. If V is irreducible then the Jacquet module VN is finite dimensional. Infact we have dim(VN) ≤ 2.

Proof. We first prove that VN contains a one-dimensional representation of A. We alreadyknow that VN is a smooth admissible as representation of A. For some compact opensubgroup AK of A, V AK

N is non zero and finite dimensional. The commutative group of finitetype A/AK acting on this finite dimensional vector space admits at least one eigenvector.Thus there exists a non zero vector v ∈ VN and a character χ : A → C× so that for everya ∈ A, πN(a)v = χ(a)v.

Now we have a non zero B-map V → χ. We derive from the Frobenius reciprocity anon zero map V → IndGB(χ) which is necessarily injective since V is irreducible. Since theJacquet functor is exact, we have an injective map VN → IndGB(χ)N . It is thus enough toprove the following statement.

Proposition 4.7.8. dim IndGB(χ)N = 2 for all character χ of A.

Proof. It amounts to prove that the space of linear functional L : IndGB(χ) → C such thatL(πχ(n)v) = L(v) for all v ∈ IndGB(χ), n ∈ N and πχ is the representation of G in IndGB(χ).Let L be such a functional which is nonzero. The space of the induced representation consistsof function f : G → C so that for all b ∈ B, we have f(bg) = χ(b)f(g). We have a linearmap

Λ : S(G)→ IndGB(χ)

defined by

Λ(φ)(g) =

∫B

φ(bg)δ−1(b)χ−1(b)db

where db is the left invariant measure on B. We are now considering the distribution

φ 7→ L(Λ(φ)).

It satisfies the property

(4.7.2) lbrn(L Λ) = δ(b)χ(b)(L Λ)

Recall the Bruhat decomposition G = Bw1B tB where Bw1B is an open subset and B isclosed. In applying 4.5.2, it is enough to prove that on Bw1B and on B there is exactly on

60

distribution satisfying (4.7.2). This is easy to see because B ×N , B acting on the left andN on the right, acts transitively on each Bruhat cell.

One important consequence of the finiteness of the Jacquet module is the existence ofWhittaker model for infinite dimensional representations.

Corollary 4.7.9. Infinite dimensional smooth admissible representations of G have Whit-taker model.

Proof. Let V be an infinite dimensional irreducible admissible representation of G. Assumethat V has no Whittaker model for some non trivial additive character ψ. Using the transitiveaction of A on the set of non trivial characters of N , we conclude that V has no Whittakermodel for any non trivial additive character of N . As S(F )-module 4.6.4, we have Vx = 0for all x 6= 0. It follows that map V → V0 is an isomorphism. We know that the fiber of Vover x = 0 is the Jacquet module VN which is finite dimensional. It follows that V is finitedimensional.

4.8. Principal series. A smooth character χ : F× → C× is a character that is trivial ona compact open subgroup of F×. Recall that F× = O×F × pZ. The character χ consists ina finite order character χ0 : O×F → C× and a complex number s well determined modulo2iπ log(p) such that χ(p) = p−s. The character χ is unitary if |z| = 1. The character is saidto be unramified if χ0 = 1.

Let χ : A → C× be a smooth character. Since A = F× × F×, χ = (χ1, χ2) where χ1, χ2

are characters of F× as above. We say χ is unramified if χ is trivial on A∩K which amountsto saying that both χ1 and χ2 are unramified. We define the principal series representationiGB(χ) as the induces representation

iGB(χ) = IndGB(χ⊗ δ1/2).

Recall that this is the space of locally constant functions f : G→ C such that

(1) for every b ∈ B, f(bg) = δ1/2B (b)χ(b)f(g),

(2) there exists a compact open subgroup K of G so that f(gk) = f(g) for all k ∈ K.

The translation of G on the right defines an action of G on iGB(χ). The second conditionmakes sure that iGB(χ) is a smooth representation.

Proposition 4.8.1. If χ is unitary the iGB(χ) is a unitary representation. More generally,the contragredient of iGB(χ) is iGB(χ−1) where χ−1 = (χ−1

1 , χ−12 ).

Proof. Let f1 ∈ iGB(χ1, χ2) and f2 ∈ iGB(χ1, χ2) then f1f2 ∈ IndGB(δ). There is now a canonicalG-invariant linear form IndGB(δ) → C. This defines a nonzero G-invariant pairing betweeniGB(χ1, χ2) and iGB(χ−1

1 , χ−12 ).

Proposition 4.8.2. Let χ = (χ1, χ2) be a character of A. Except if χ′ = (χ1, χ2) orχ′ = (χ2, χ1), there is no nonzero intertwining operators from iGB(χ) to iGB(χ′).

Proof. Assume that there is a nonzero intertwining operator iGB(χ) → iGB(χ′). It induces anon zero G-invariant bilinear form iGB(χ) × iGB(χ′−1) → C. We have canonical projectionsΛχ : S(G)→ iGB(χ) defined by

Λχ(φ)(g) =

∫B

φ(bg)δ−1/2(b)χ(b)db

61

where db is the left invariant measure on B. Similarly, there is a G-equivariant canonicalmaps

Λχ′−1 : S(G)→ iGB(χ′−1

).

It follows that there is a canonical linear form

c : S(G×G)→ C

defined by the formulac(φ1 ⊗ φ2) = 〈Λχ(φ1),Λχ′−1(φ2)〉

using the pairing between iGB(χ) and iGB(χ′−1). This is a G-invariant with respect to thediagonal action of G by right translation of G on G×G. There exists a unique linear formb : S(G)→ C such that

c(φ1 ⊗ φ2) = b(φ1 ∗ φ′2)

where φ′2(g) = φ2(g−1). Now from the construction we have

Λχ(lb1φ1) = δ−1/2(b1)χ(b1)Λχ(φ1).

Similarly, we haveΛχ′−1(lb2φ2) = δ−1/2(b2)χ′

−1(b2)Λχ′−1(φ2).

We have

(4.8.1) (lb1φ1) ∗ (lb2φ2)′ = lb1rb2(φ1 ∗ φ′2).

It follows taht for all φ ∈ S(G), we have

b(lb1rb2φ) = δ−1/2(b1)χ(b1)δ−1/2(b2)χ′−1

(b2)Λ(φ).

We use again the Bruhat decomposition G = Bw1B ∪ B. On each Bruhat double cosetthere is at most one distribution satisfying the transformation property (4.8.1).

On the big cell Bw1B, there is a non zero distribution satisfying (4.8.1) if and only if therestriction of the character

(4.8.2) (b1, b2) 7→ δ−1/2(b1)χ(b1)δ−1/2(b2)χ′−1

(b2)

to the stabilizer (b1, b2) | lb1rb2w1 = w1. The stabilizer consists in pairs (a1, a2) witha1, a2 ∈ A such that a1 = w1a2w1. In that case δ−1/2(a1)δ−1/2(a2) = 1 so that the restrictionof the above character to the stabilizer is trivial if and only if χ′ = w1χ i.e if χ = (χ1, χ2)then χ′ = (χ2, χ1).

On the small cell B, there is a non zero distribution satisfying (4.8.1) if and only if χ = χ′.Here we have to take the modulus character δ of B in to account.

Proposition 4.8.3. The representation iGB(χ) admits an invariant one-dimensional subspaceif and only if χ1χ

−12 (y) = |y|−1 for all y ∈ F×. It admits an invariant one-dimensional

quotient if and only if χ1χ−12 (y) = |y| for all y ∈ F×.

Proof. Consider one-dimensional representation of G which is a subspace of iGB(χ). Thecommutator group SL(2, F ) acts trivially on the one dimensional subspace. It implies thatthe restriction of the character δ1/2χ to the one dimensional torus[

y 00 y−1

]is trivial. In other words, we have χ1χ

−12 (y) = |y|−1.

62

Now iGB(χ) admits a one-dimensional quotient if and only if its contragredient iGB(χ−1)admits a one-dimensional subspace.

If iGB(G) admits a one-dimensional representation as subspace, then the quotient by thissubspace is called special representation. For every character χ2 : F× → C×, we denote σ(χ2)the special representation given as the irreducible quotient of iGB(χ1, χ2) where χ1χ

−12 (y) =

|y|−1.

Proposition 4.8.4. Except the above mentioned cases χ1χ−12 (y) = |y|±1, the representation

iGB(χ) is irreducible. Special representations are irreducible.

Proof. Except in the cases χ1χ−12 (y) = |y|±1, iGB(χ) do not have one-dimensional (so finite

dimensional) subspace or quotient. If it is not irreducible, there exists an exact sequence

0→ V ′ → iGB(χ)→ V ′′ → 0

where both V ′ and V ′′ are infinite dimensional. It follows that V ′ and V ′′ admits at least oneWhittaker functional after 4.7.9 and therefore iGB(χ) admits at least two linearly independentWhittaker functionals which would contradict the following statement

Proposition 4.8.5. For every nontrivial additive character ψ, dim(iGB(χ))N,ψ = 1.

Proof. The proof is similar to 4.8.2 and is based on the Bruhat decomposition.

Theorem 4.8.6. Let V be an irreducible admissible representation of G. If V is a principalseries representation iGB(χ) then dim(VN) = 2 and A acts on VN through δ1/2χ and δ1/2w1(χ).If V is a special representation the dim(VN) = 1. Otherwise VN = 0 i.e. V is supercuspidalrepresentation.

Proof. If the irreducible representation V ' iGB(χ) is a principal series representation thenV ' iGB(w1χ). The isomorphism V ' iGB(χ) induces a nonzero A-map VN → δ1/2χ. Similarly,we have a nonzero A-map VN → δ1/2w1(χ). This implies that dim(VN) ≥ 2. If dim(VN > 2)then a character χ′ /∈ χ,w1(χ) such that there exists a nonzero intertwining operatorV ' iGB(χ)→ iGB(χ′) which would contradict 4.8.2.

Proposition 4.8.7. The space of K(1)-fixed vectors iGB(χ) is of dimension no greater thanone. It is of dimension one if and only if χ1 and χ2 are unramified characters.

Proof. It follows from the Iwasawa decomposition 4.1.2, a function which is K(1)-right in-variant and which transform on the left by the character χ is completely determined by itsvalue at the origin. At the origin, the two conditions can reconciled if and only if χ is trivialon B ∩K in other words if χ is an unramified character. In that case dim iGB(χ)K(1) = 1.

4.9. Kirillov model. Let ψ : F → C× denote a non trivial additive character. Then allcharacters of F must be of the form ψa : x 7→ ψ(ax) for some a ∈ F×.

Let (π, V ) a smooth irreducible representation of G that admits a ψ-Whittaker functionalΛψ : V → C. For every v ∈ V , the corresponding Whittaker function.Wv ∈ IndGN(ψ) is givenby g 7→ Λ(π(g)v). The construction of the Kirillov model consist in restricting Whittakerfunction to the subgroup F× of matrices of the form

a =

[a 00 1

]63

Proposition 4.9.1. Assume that V is infinite dimensional. If v 6= 0, then there exists

a ∈ F× so that Wv

[a 00 1

]6= 0.

Proof. Assume that Wv(a) = 0 for all a ∈ F×. This implies that Λψa(v) = 0 because

v 7→ Λψ(π

[a 00 1

]v)

is a ψa-Whittaker functional. In other words, the image of v in Va vanishes for every a ∈ F×.Here we are considering V as a smooth S(F )-module as in 4.6.4. For every x ∈ F , we have

(4.9.1) π

[1 x0 1

]v = v.

In fact, if v′ denote the difference of these two vectors, then the image of v′ in Va vanishesfor every a ∈ F so that v′ = 0.

The equation (4.9.1) being established for all x ∈ F , v is a N -invariant vector. Byassumption, v is also invariant under an open compact subgroup K and in particular bya K ∩ N− where N− is the group of unipotent lower matrices. Now v is stablized by theSL(2, F ) because this group is generated by any two non trivial unipotent matrices, oneupper and the other lower triangular. This implies that V is finite dimensional.

For every v, let φv : F× → C denote the function

φv(a) = Wv

[a 00 1

].

We denote Kr(V ) the subspace of complex valued functions of F× image of the applicationv 7→ φv. Since this application is injective, V = Kr(V ) so that Kr(V ) also affords a smoothirreducible representation of G. This is the Kirillov model of V . It is easy to write down theaction of upper triangular matrices in Kr(V )

(4.9.2) π

[a 00 1

]φv(y) = φv(ay) and π

[1 b0 1

]φv(y) = ψ(by)φv(y)

The equations (4.9.2) impose restrictions on φv associated to a vector v. Since v is fixedby some compact subgroup K, φv must be invariant under the translation by a compactsubgroup of F× by the first equation. Since v is invariant under N ∩K, the second equationimplies φv(y) = 0 if the character b 7→ ψ(by) is non trivial on N ∩K, in other words thereexists C > 0 such that φ(y) = 0 for all |y| > C.

Proposition 4.9.2. For every v ∈ V , the function φv : F× → C is locally constant and itssupport is contained in a compact subset of F . Moreover, if v has zero image in the Jacquetmodule VN , then the function φv has compact support

Proof. The first statement has just been proved. For the second statement, the kernel V (N)of V 7→ VN is generated by vectors of the form[

1 b0 1

]v − v.

The associated function φ(y) vanishes for |y| small enough so that ψ(by) = 1. 64

Theorem 4.9.3. Let (π, V ) be an infinite-dimensional irreducible smooth representation ofG. Let identify V with its Kirillov model. Then the kernel V (N) of the projection from Vto its Jacquet module VN is exactly C∞c (F×).

Proof. V (N) is a non-zero B1-invariant subspace of C∞c (F×). It is enough to prove thatC∞c (F×) as representation of B1 is irreducible.

From the description of Jacquet module of V . we deduce the following proposition.

Proposition 4.9.4. Let us denote by πKr the action of the group

(4.9.3) B1 =

[a b0 1

]| a ∈ F×, b ∈ F

acts on C∞c (F×) by the equation (4.9.2). Then (πKr, C

∞c ) is an irreducible representation.

Proof. Let φ ∈ C∞c (F×). The action of F on φ according to (4.9.2) gives rise to an action ofthe algebra C∞c (F ). For every f ∈ C∞c (F ), we have

πKr(f)(φ)(y) = f(y)φ(y).

The irreducibility of πKr can be easily deduced from this equation.

Theorem 4.9.5. Let (π, V ) = iGB(χ1, χ2) be an irreducible principal series i.e. χ1χ−12 (t) is

not |y| or |y|−1.

(1) Assume taht χ1 6= χ2. Then the space of Kirillov model of V consists of the functionon F× that are locally constant, that vanish for large values of |y| and

(4.9.4) φ(y) = c1|y|1/2χ1(y) + c2|y|1/2χ2(y)

for for all |y| small and for some constants c1, c2.(2) Assume that χ1 = χ2 = χ. The space of Kirillov model of V consists of the function

on F× that are locally constant, that vanish for large values of |y| and

(4.9.5) φ(y) = c1|y|1/2χ(y) + c2val(y)|y|1/2χ(y)

for for all |y| small and for some constants c1, c2.

4.10. Local functional equation. We define local L factor of an irreducible representationas follows. If π = iGB(χ1, χ2) is a principal series representation, we put

L(s, π) = (1− α1q−s)−1(1− α2q

−s)−1

where αi = χi($) if χi is unramified and αi = 0 if χi is ramified. If π = σ(χ2) is a specialrepresentation then L(s, π) = (1− α2q

−s)−1 where again α2 = χ2($) if χi is unramified andα2 = 0 if χ2 is ramified.

Proposition 4.10.1. Let (π, V ) be an infinite dimensional irreducible smooth representationof G. If f is an element in the space of its Kirillov model, consider the interal

(4.10.1) Z(s, f) =

∫F×

f(y)|y|s−1/2d×y

This integral is convergent for <(s) >> 0 has meromoprphic continuation to all s ∈ C.There exists a unique polynomial L(π, s)−1 ∈ C[q−s] in the variable q−s with free coefficientone such that for every f ∈ V , there exists p(s, f) such that

(4.10.2) Z(s, f) = p(s, f)L(s, π).65

Moreover, one can choose f0 so that Z(s, f0) = L(s, π) i.e. p(s, f0) = 1.

Proof. Suppose that (π, V ) = iGB(χ) is a principal series with χ = (χ1, χ2). For every functionf : F× → C in the Kirillov model of π, there exist c1, c2 ∈ C such that

f(y) = c1|y|1/2χ1(y) + c2|y|1/2χ2(y)

for all t ∈ F× with |y| small.Let f0 denote the locally constant on F× supported by OF defined by

f0(y) = c1t1/2χ1(y) + c2t

1/2χ2(y).

Then f − f0 is a locally constant function on F× with compact support. The zeta integralZ(s, f − f0) is then a polynomial function on q±s.

The zeta integral

Z(s, f0) = c1

∫OF

χ1(y)|y|sd×y + c2

∫OF

χ2(y)|y|sd×y.

Both summands are absolutely convergent for <(s) >> 0. Assume <(s) >> 0. If χ1 : F× →C is a ramified character then ∫

$rOFχ1(y)|y|sd×y = 0

for all integer r. It follows that the integral∫OF

χ1(y)|y|−sd×y = 0.

If χ1 is an unramified character with χ1($) = α1 then we have∫OF

χ1(y)|y|−sd×y =∞∑n=0

αn1p−sn

= (1− α1p−s)−1

Thus

Z(s, f0) = c1(1− α1p−s)−1 + c2(1− α2p

−s)−1

so thatZ(s, f0)

L(s, π)= c1(1− α2p

−s) + c2(1− α1p−s)

is a polynomial function on p−s. Here L(s, π) = (1 − α1p−s)−1(1 − α2p

−s)−1. Notice thatsince α1 6= α2, the scalars c1, c2 can be choosen such that c1(1− α2p

−s) + c2(1− α1p−s) = 1.

In this case we have

p(s, f0) = 1.

In the above argument, we only use the asymptotic expansion of f around 0 based onthe Jacquet module of iGB(χ). The same argument works with special representation andsupercuspidal representation as well. In the supercuspidal case L(s, π) = 1 and we canchoose f0 = 1O×F

. 66

More generally, for all characters ξ : F× → C×, we can define

(4.10.3) Z(s, f, ξ) =

∫F×

f(y)χ(y)|y|s−1/2d×y

as well as the factor L(s, π, χ) so that

(4.10.4) Z(s, f, χ) = p(s, f, χ)L(s, π, χ).

Proposition 4.10.2. Let (π, V ) be an irreducible smooth representation of G and let χ be acharacter of F×. Then for all but at most two values of s modulo 2πi/ log(q), the dimensionof the space of linear functionals Λ : V → C satisfying

(4.10.5) L

(π

[y 00 1

]v

)= χ(y)|y|sL(v)

is equal to one.

Proof. Let L1, L2 denote two linearly independent linear form satisfying the equation 4.10.5.Since V (N) = C∞c (F×) and the space of χ-eigendistribution on F× is one-dimensional, L1

and L2 are proportional on V (N). There exist c1, c2 not both zero so that c1L1 +c2L2 factorsthrough VN . But dimVN ≤ 2, and there are at most two characters of F× that appears in VN .If y 7→ χ(y)|y|s is not one of these characters, the space of functionals Λ satisfying (4.10.5)is of dimension not greater than one. By the above proposition, we know that f 7→ pχ(f, s)is a non zero element of this space. Therefore, its dimension is at least one.

Theorem 4.10.3. Let (π, V ) be an infinite-dimensional irreducible smooth representationof G with central character ω. Let χ be a character of F×. We identify V with its Kirillovmodel. There exists a meromorphic function γ(s, π, ξ, ψ) such that for all f ∈ V we have

(4.10.6) Z(1− s, π(w1)f, ω−1χ−1) = γ(s, π, χ, ψ)Z(s, f, χ)

where w1 is the matrix

(4.10.7) w1 =

[0 1−1 0

]Proof. Both fΛ 7→ L1(f) = Z(s, f, χ) and f 7→ L2(f) = Z(1− s, π(w1)f, ω−1χ−1) are linearfunctional on V that satisfy

(4.10.8) L

(π

[a 00 1

]f

)= χ(a)−1|a|−s+1/2L(f).

First we check this property for L1.

L1(π

[a 00 1

]f) =

∫F×

f(ay)χ(y)|y|s−1/2d×y

= χ(a)−1|a|−s+1/2L(f).67

Now we will check this property for L2.

L2(π

[a 00 1

]f) =

∫F×

(π(w1)π

[a 00 1

]f

)(y) (ω−1χ−1)(y)|y|−s+1/2d×y

=

∫F×

ω(a)

(π

[a−1 00 1

]π(w1)f

)(y)ω−1(y)χ−1(y)|y|−s+1/2d×y

=

∫F×

(π(w1)f)(a−1y)ω−1(a−1y)χ−1(y)|y|−s+1/2d×y

= χ(a)−1|a|−s+1/2L2(f)

For all s except two at complex values, the space of such linear functionals is one dimen-sional. There exists a proportionality constant γ(s, π, ξ, ψ) between the two linear form whichdepend holomorphically on s. Both Z(1−s, π(w1)f, ω−1χ−1) and Z(s, f, χ) are meromorphicfunction on s, then so is γ(s, π, χ, ψ).

Theorem 4.10.4. There exists an invertible holomorphic function σ(s, π, ψ) such that

γ(s, π, χ, ψ) =L(1− s, π, χ−1)

ε(s, π, χ, ψ)L(s, π, χ)

where π is the contragredient representation of π.

Proof. By definition, we have

γ(s, π, χ, ψ) =Z(1− s, π(w1)f, ω−1χ−1)

Z(s, f, χ)

for all functions f : F× → C in the Kirillov model of π. Let consider the function f : F× → Cgiven by

f(y) = π(w1)f(y)ω−1(y).

We observe that f is an element of the Kirillov model of the contragrendient representationπ of π. If π = iGB(χ1, χ2) is an unramified principal series, there exists c1, c2 ∈ C such that

π(w1)f(y) = δ1/2(y)(c1χ1(y) + c2χ2(y))

for small |y|. We have ω(y) = χ1χ2(y) so that

f(y) = δ1/2(y)(c1χ−11 (y) + c2χ

−12 (y))

belongs to the Kirillov model of the contragredient representation π = iGB(χ−11 , χ−1

2 ). The

same argument works in other cases. In fact f 7→ f induces a bijection from the Kirillovmodel of π on the Kirillov model of π.

We haveZ(1− s, π(w1)f, ω−1χ−1) = p(s, f , χ−1)L(1− s, π, χ−1)

where p(s, f , χ−1) is some polynomial in q±s which is equal one for some f .We have

γ(s, π, χ, ψ) =L(1− s, π, χ−1)

ε(s, π, χ, ψ)L(s, π, χ)

with

ε(s, π, χ, ψ) =p(s, f , χ−1)

p(s, f, χ).

68

We exploit the fact the ε does not depend on f to prove that it is an invertible holomorphicfunction. We can choose f such that p(s, f, χ) = 1 i.e. ε is a holomorphic function on s. We

can also choose f such that p(s, f , χ−1) = 1 which implies that ε is inverse to a holomorphicfunction. So ε is an invertible holomorphic function.

4.11. Formal Mellin transform.

5. Automorphic representations on adelic groups

In this chapter A will denote the ring of adeles of Q. Let Afin denote the ring of finiteadeles. We have A = Afin × R. More generally, if F is a global field, we will denote AF itsring of adeles. We have AF = Afin

F × A∞F .In this chapter, we will use the letter G for the group GL2.

5.1. Adeles and ideles. The absolute values of the field of rational numbers Q is eitherarchimedean or non archimedean, thus associated a prime number p or archimedean. Recallthat the p-adic absolute value of m/n with m,n ∈ Z is |m/n| = p−ordp(m)+ordp(n) where ordpof an integer is the order of the largest power of p dividing it. While the completion of Qwith respect to the archimedean absolute value is the field R of real numbers, the completionQp of Q with the p-adic absolute value is the field of p-adic numbers. It is the fraction field ofthe ring Zp of p-adic integers that is the completion of Z with respect to the p-adic absolutevalue.

The ring A of adeles of Q is the restricted product

A =∏

p∈PQp × R

whose elements are xA = (xp, x)p∈P with x ∈ R, xP ∈ Qp for all p and xp ∈ Zp for all but

finitely many p ∈ P . We also write A = Afin × R where Afin =∏

p∈PQp is called the ring

of finite adeles. We equip A and Afin with the Tychonoff product topology which makes itlocally compact. The compact subring

∏p∈P Zp is the profinite completion Z of the ring of

integers Z. The field of rational numbers Q is contained in A as a discrete subring.

Lemma 5.1.1. We have Q ∩∏

p∈P Zp = Z and Q\A/∏

p∈P Zp = R/Z. In particular Q\Ais a compact group.

Proof. A rational number m/n with (m,n) = 1 that is p-integral for all prime p as n has

no prime factors. Thus Q ∩ Z = Z. A coset in Afin/∏

p∈P Zp can be represented by a finite

collection of fractions (mp/prp)p∈S indexed by a finite set S of prime numbers. The sum

m/n =∑

p∈Σ mp/prp satisfies the property that m/n ≡ mp/p

rp for all p. This implies that

Afin = Q +∏

p∈P Zp. It also follows that Q\A/∏

p∈P Zp = R/Z.

The group A× of ideles of Q is the restricted product

A× =∏

p∈PQ×p × R×

whose elements are xA = (xp, x)p∈P with x ∈ R×, xP ∈ Q×p for all p and xp ∈ Z×p for all but

finitely many p ∈ P . We also write A× = Afin××R× where Afin× is the group of finite ideles.69

For xA ∈ A×, let us consider the norm |xA| =∏

p∈P |xp|p|x∞|∞ the infinite product is well

defined because all but finitely many of its terms is equal to one. Let A1 be the subgroup ofelements of norm one in A×.

Lemma 5.1.2. We have Q× ∩∏

p∈P Z×p = ±1 and Q×\A×/∏

p∈P Z×p = R×+. The group

of norm one ideles A1 contains Q× as a cocompact subgroup i.e. the quotient Q×\A1 is acompact group.

Proof. A rational number that is p-integral and whose inverse is also p-integral must be ±1.For every prime p, there is an isomorphism Q×p /Z×p = Z given by the p-adic valuation. The

quotient Afin×/∏

p∈P Zp is given by a collection of integers (rp)p∈P that vanish except for

finitely many of them. Since this class can be represented by the rational number∏

p∈P prp ,

we have Q×\Afin×/∏

p∈P Zp = 1. It follows that Q×\A×/∏

p∈P Z×p = R×/±1 that is

isomorphic to R×+ by the application x 7→ |x|.In order to prove Q× ⊂ A1, it is enough to prove the product formula

|x|∞∏p∈P

|x|p = 1

for all x ∈ Z − 0. But this follows immediately from the decomposition in prime factorsx = ±

∏prp . We have |x|p = p−rp for all prime p and |x|∞ =

∏prp . According to the

isomorphism Q×\A×/∏

p∈P Z×p = R×+ above, we have Q×\A1/∏

p∈P Z×p = 1. This implies

the compacity of Q×\A1.

For every number field F , the ring of adeles AF is defined in a similar manner. An absolutevalue of F is either archimedean or non archimedean. For an archimedean absolute value v,an infinite place, the completion Fv is either the field of real numbers C or the field of realnumbers R. A non archimedean absolute value v, a finite place, is up to normalization is anextension of the p-adic valuation on Q; the v-adic completion Fv is a finite extension of Qp.We will denote by Ov its ring of integers. The ring of adeles AF is the restricted product ofall those local fields Fv whose elements are (xv) with xv ∈ Fv for all v and xv ∈ Ov for allbut finitely many v. Again, AF = F∞×AF,fin where F∞ is the product of the completion atinfinite places and the ring of finite adeles AF,fin is the restricted product of the completionsat all finite places. We denote Ofin =

∏vOv the product over all the finite places v of F of

the valuation ring Ov in Fv. Let OF denote the ring of integers of F .

Proposition 5.1.3. The quotient F\AF is a compact group.

Proof. We have F∩Ofin = OF , the ring of integers of F . It follows that F\AF/Ofin = OF\F∞.Here F∞ = F⊗QR is a r-dimensional real vector space that containsOF as a complete lattice.The quotient OF\F∞ is thus compact.

Proposition 5.1.4. A×F,1 contains F× as a subgroup and the quotient F×\A×F,1 is a compactgroup.

Proof. We will see that F×\A×F,1/O×fin is a compact group. Consider the projection on the

finite ideles

prfin : F×\A×F,1/O×fin → F×\A×F,fin/O

×fin.

70

The latter group is the finite by the theorem of finiteness of ideal classes. The kernel of prfin

is (F× ∩ O×fin)\F×∞. We have F× ∩ O×fin = O×F , the group of units of OF . The compacity ofthe quotient O×F \F 1

∞ is a formulation of the Dirichlet unit theorem.

The group F×\A×F is called the group of ideles classes. This is a locally compact groupequipped with an absolute value

F×\A×F → R+

whose kernel F×\A1F is a compact group.

5.2. The case of function fields. Let F be the field of rational functions on a smoothprojective curve connected C defined over a finite field k. The places of F are the closedpoints of C. Of course, there is no archimedean place. The ring of adeles AF contains adiscrete subring F and a compact subring OAF =

∏v∈|C|Ov where Ov is the completion of

the local ring OC,v of C at v.

Proposition 5.2.1. We have F ∩ OAF = H0(C,OC) and F\AF/OAF = H1(C,OC). Inparticular, they are both finite dimensional k-vector spaces.

Proposition 5.2.2. The quotient A×F/O×AF

can be identified with the group of divisors of C,

the double quotient F×\A×F/O×AF

with the group PicC of isomorphism classes of line bundle

on C. The degree of a line bundle is a homomorphism deg : F×\A×F/O×AF→ Z whose kernel,

the group Pic0C of line bundles of degree 0, is finite.

The absolute value on F×\A×F is given by |x| = q−deg(x) where q is the cardinal of the basefield k. Its kernel is the compact group F×\A1

F that is an extension of the finite group Pic0C

by the compact group O×AF . The choice of a line bundle of degree one provides a splitting

PicC = Pic0C × Z.

5.3. Strong approximation theorem.

Theorem 5.3.1. SL2(Q) is dense in SL2(Afin).

Proof. Let M denote the closure of SL2(Q) in SL2(Afin). We first prove that M containsSL(Qp) embedded as the p-component of SL2(Afin). It is enough to prove that M contains thesubgroups N(Qp) and N−(Qp) since these subgroups generate SL2(Qp). But this statementderives from the density of Q in Afin.

Let S be a finite set of primes. It is enough to prove that M contains∏

p∈S SL2(Qp) ×∏p/∈S SL2(Zp) because by enlarging S, these groups cover SL2(Afin). SinceM contains already

SL2(Qp) for p ∈ S, it is enough to prove that M contains the profinite group∏

p/∈S SL2(Zp).This is equivalent to prove that M ∩

∏p/∈S SL2(Zp) maps onto the finite quotient SL2(Z/nZ)

for every integer n prime to S. But this follows from the fact SL2(Zp) ⊂ M for all p|n thatwe already know.

Corollary 5.3.2. For every compact open subgroup K0 of SL(Afin), let us denote Γ0 =SL2(Q)∩K0.The embedding of SL2(R) as the infinite component of SL2(A) induces a home-omorphism

Γ\SL2(R)→ SL2(Q)\SL2(A)/K0.

71

Proof. We first prove that the map

SL2(R)→ SL2(Q)\SL2(A)/K0

is surjective. An element of

SL2(A)/K0 = (SL2(Afin)/K0)× SL2(R).

can be written as x = (xfin, g∞) with xfin ∈ SL2(Afin)/K0 and g∞ ∈ SL2(R). The densityof SL2(Q) in SL2(Afin) implies that there exists γ ∈ SL2(Q) such that whose image inSL2(Afin)/K0 is xfin. The transformation formula γ−1(xfin, g∞) = (1, γ−1g∞) implies thedesired surjectivity. Now let γ1, γ2 ∈ SL2(Q) having the same image xfin ∈ SL2(Afin)/K0

then γ2 = γ1γ whereγ ∈ SL2(Q) ∩K0 = Γ0.

It follows that the mapΓ\SL2(R)→ SL2(Q)\SL2(A)/K0

is a homeomorphism.

For every positive integer N let K0(N) =∏

pK0(N)p where K0(N)p be the subgroup of

GL2(Zp) of matrix with congruent to an upper triangular matrices modulo N . We have

Γ0(N) = SL2(Q) ∩K0(N).

Proposition 5.3.3. We have a homeomorphism

Γ0(N)\SL2(R) ' Z(A)GL2(Q)\GL2(A)/K0(N).

Proof. This is the combination of the above corollary and the fact that Q has class numberone.

5.4. Automorphic representations and automorphic forms.

Proposition 5.4.1. The quotient space G(Q)Z(A)\G(A) has finite measure.

For every unitary character ω : F×\A× → C×, we consider the space L2(G(F )\G(A), ω)of G(F )-functions φ on G(A), that transform by the character ω with respect to the actionof the center Z, and whose module |φ| is a square integrable function on G(Q)Z(A)\G(A).

A function φ is said to be cuspidal if∫N(F )\N(A)

φ(ng)dn = 0

for all g ∈ G(A). The subspace of cuspidal functions L2cusp(G(F )\G(A), ω) is a closed sub-

space of L2(G(F )\G(A), ω). Both L2(G(F )\G(A), ω) and L2cusp(G(F )\G(A), ω) are Hilbert

representations of G(A).

Theorem 5.4.2. The space L2cusp(G(F )\G(A), ω) decomposes as a Hilbert direct sum of

irreducible invariant subspaces.

Let A(G,ω) denote the space of smooth functions φ : G(A)→ C such that

(1) φ transforms under the action of Z(A) according to ω,(2) φ is K-finite with respect to any compact subgroup K = KfinK∞ where Kfin is a

compact open subgroup of G(Afin) and Kfin is the maximal compact subgroup ofSL2(R),

72

(3) φ is Z(U(g))-finite(4) φ has moderate growth i.e. |φ(g)| ≤ C||g||N for some constant C ∈ R+ and N ∈ N.

Such a function φ is called an automorphic form. The K-finiteness implies that there existsa compact open subgroup K0 of G(Afin). If ω = 1 then φ can be seen as an automorphicfunction

φ : Γ0\SL2(R)→ C.As we have already seen in the framework on automorphic representations on real groups,a L2-automorphic functions that are K-finite and Z(U(g))-finite are analytic and will auto-matically have moderate growth. We will also consider the space

Acusp(G,ω) = A(G) ∩ L2cusp(G(F )\G(A), ω)

of cuspidal automorphic functions. The spaces A(G,ω) and Acusp(G,ω) are smooth G(Afin)-representations and (g, K∞)-modules.

Definition 5.4.3. An irreducible admissible G(A)-module is a restricted tensor product

π =⊕p

πp

with p runs over all places of Q where

(1) (π∞, V∞) is an irreducible admissible (g, K)-module(2) for all finite place p, (πp, Vp) is an irreducible admissible representation of G(Qp),

(3) for all but finitely many p, πp is unramified i.e dim(VKpp ) = 1.

By restricted tensor products we mean the space of vector of the form

v =⊗p

vp

where vp ∈ V Kpp for almost all prime p.

Theorem 5.4.4. Let π be an irreducible G(A)-invariant subspace of L2(G(F )\G(A), ω).The π ∩ A(G,ω) is an admissible G(A)-module in the above sense.

5.5. Fourier expansion and Whittaker models.

Theorem 5.5.1. Let (π, V ) be an automorphic cuspidal representation of G with

V ⊂ A0(G(F )\G(A), ω)

where ω : F×\A× → C× is a unitary character of the center. If φ ∈ V and g ∈ G(A), let

Wφ(g) =

∫F\A

φ

([1 x0 1

]g

)ψ(−x)dx.

The space WV of functions Wφ is a Whittaker model of π. We have the Fourier expansion

(5.5.1) φ(g) =∑a∈F×

Wφ

([a 00 1

]g

).

73

Proof. The function f : A→ C

f(x) = φ

([1 x0 1

]g

)is smooth and periodic with respect to the discrete cocompact subgroup F . It admits theFourier expansion

f(x) =∑a∈F

c(a)ψ(ax)

since F can be identified with the Pontryagin dual of F\A. The coefficient c(a) is given bythe integral

c(a) =

∫F\A

φ

([1 x0 1

])ψ(−ax)dx.

Since φ is cuspidal c(0) = 0. For a ∈ F×, we have

Wφ

([a 00 1

]g

)=

∫F\A

φ

([1 x0 1

] [a 00 1

]g

)ψ(−x)dx

=

∫F\A

φ

([a 00 1

] [1 a−1x0 1

]g

)ψ(−x)dx

=

∫F\A

φ

([1 x0 1

])ψ(−ax)dx.

= c(a)

where we have used the invariance property of φ with respect to the translation of

[a 00 1

].

5.6. Multiplicity one. We will prove the strong multiplicity one theorem of Piatetski-Shapiro.

Theorem 5.6.1. Let (π, V ) and (π′, V ′) be two automorphic cuspidal representations of Gi.e.

V, V ′ ⊂ Acusp(G(F )\G(A), ω)

for some unitary character ω of the center. Assume that πv ' π′v for all infinite places andfor all but finitely many finite places. Then V = V ′.

Proof. If πv ' π′v for all v then they have the same model of Whittaker. The equality V = V ′

follows from (5.5.1).Now we suppose that πv ' π′v for all infinite places and for all but finitely many finite

places. Consider their Whittaker model⊕

v Wh(Vv) and⊕

v Wh(V ′v) with Wh(Vv) = Wh(V ′v)for almost all places including the infinite place. We choose Whittaker functions W =

⊗vWv

and W ′ =⊗

vW′v as follows

(1) W∞ = W ′∞ is an eigenvector of K∞

(2) for all p such that Wh(Vp) = Wh(V ′p) is unramified we choose Wv = W ′v being a

normalized Kv-invariant vector,(3) for finitely remaining v we choose Wv and W ′

v such that they have the same Kirillovfunction F×v → C which is a smooth compactly supported function.

74

The last condition can be made possible because Kirillov model of any infinite dimensionalrepresentation contains C∞c (F×v ).

By 5.5.1 the series

φ(g) =∑a∈F×

W

([a 00 1

]g

)and

φ′(g) =∑a∈F×

W ′([a 00 1

]g

)are convergent to nonzero function φ, φ′ : G(F )\G(A) → C. By our choice of W and W ′,they agree on B(A). They are both invariant on the right by some K0 where K0 is somecompact open subgroup of G(Afin) and is a eigenvector of K∞ with the same eigenvalue. Itfollows that φ and φ′ agree over

B(A)K0K∞.

The strong approximation theorem implies that

G(F )B(A)K0K∞ = G(A).

Since both φ and φ′ are G(F )-invariant on the left, it follows φ = φ′.The irreducible representations V and V ′ sharing a nonzero vector, are the equal.

5.7. Hecke theory from the Jacquet-Langlands point of view.

Theorem 5.7.1. Let π =⊗

v πv be an irreducible admissible G(A)-module that occurs inAcusp(G,ω). Define L(s, π) =

∏v L(s, πv). Then

(1) L(s, π) and L(s, π) converge in a right half plane and can be holomorphically continuedto C.

(2) They are bounded in any finite vertical strip.(3) L(s, π) = ε(s, π)L(1 − s, π) with ε(s, π) =

∏ε(s, πv, ψv) for any nontrivial additive

character ψ : F\A→ C.

Proof. Let V denote the space of the automorphic cuspidal representation π. For each φ ∈ V ,we consider the integral

(5.7.1) Z(s, φ) =

∫F×\A×

φ

([a 00 1

])|a|s−1/2d×a.

Since φ is rapidly decreasing when |a| → ∞, this integral is absolutely convergent for alls ∈ C and defines a holomorphic function in s which is bounded on every vertical strip.

The Fourier expansion of φ

φ(g) =∑a∈F×

Wφ

([a 00 1

]g

)allows us to unfold this integral as

(5.7.2) I(s, φ) =

∫A×Wφ

([a 00 1

])|a|s−1/2d×a.

75

If φ is a decomposable vector, its Whittaker function is an infinite product

Wφ(g) =∏v

Wφ,v(gv)

for all g ∈ G(A). Here for all most all v, Wφ,v is the normalised Whittaker function andgv ∈ G(Ov) so that Wφ,v(gv) = 1. For <(s) ≥ 0, (5.7.2) is an Eulerian product∏

v

∫F×v

Wφ,v

([av 00 1

])|av|s−1/2d×av.

Thus for <(s) ≥ 0, we have

Z(s, φ) =∏v

Z(s,Wφ,v).

We have Z(s,Wφ,v) = L(s, πv)p(s,WW,φ,v) where p(s,WW,φ,v) is a polynomial which is equalto one if Wφ,v is the normalized unramified Whittaker function. It follows that

Z(s, φ) = L(s, π)∏

v ramified

p(s,Wφ,v)

and thus L(s, πv) admit a meromorphic continuation. At the ramified place, it is possibleto choose Wφ,v so that p(s,Wφ,v) = 1 which implies that L(s, π) is holomorphic if π is aautomorphic cuspidal representation.

Using the automorphy of φ, we have another development of Z(φ, s) as Euler product

φ

([a 00 1

])= φ

([0 1−1 0

] [a 00 1

])= ω(a)φ

([a−1 00 1

] [0 1−1 0

])Thus

Z(φ, s) =

∫F×\A×F

φ

([a−1 00 1

] [0 1−1 0

])ω(a)|a|s−1/2d×a

=

∫F×\A×F

(π(w1)φ)

([a 00 1

])ω−1(a)|a|1/2−sd×a

after the change of variable a 7→ a−1. For <(s) << 0, we have

Z(φ, s) =∏v

Z(1− s, ω−1(π(w1)Wφ,v))

=∏v

L(1− s, πv)ε(s, πv, ψv)p(s,Wφ,v)

according to the local functional equation. Here ε(s, πv, ψv) and p(s,Wφ,v) are equal to 1for almost all v. This implies that L(s, π) has meromorphic continuation to all the complexplane and that we have the functional equation

L(1− s, π)ε(s, π) = L(s, π)

where ε is an exponential function. 76

For all character χ : F×\A× → C×, the same argument prove that for all automorphiccuspidal representation π the L-function L(s, π, χ) has holomorphic continuation as well asL(1− s, π, χ−1) and we have the functional equation

L(s, π, χ) = ε(s, π, χ)L(1− s, π, χ−1).

Moreover, the argument can be reversed to prove the converse theorem.

Theorem 5.7.2. Let π be an irreducible admissible G(A)-module so that L(s, π, χ) is holo-morphic and satisfies the above functional equation for all character χ. Then π is automor-phic cuspidal.

77

Appendix A. Review on compact Riemann surfaces

Let X be a compact Riemann surface. By GAGA theorem, X is a smooth projective curve.Concretely, this means that the field of meromorphic functions F on X is finite extension ofthe field of fractions C(t) of the polynomial ring C[t]. In fact any non constant meromorphicfunction f ∈ F defines a finite morphism f : X → P1

C and by assigning t 7→ f , we make Fa finite extension of C(t).

A.1. Divisors. The group of divisors Div(X) is the free abelian group with basis the set ofpoints x ∈ X. Its elements are finite linear combinations

∑i dixi with di ∈ Z and x ∈ X.

The application∑

i dixi 7→∑

i di defines a homomorphism deg : Div(X) → Z. We denoteDiv0(X) the group of divisors of degree 0. A divisor D ∈ Div(X) is said to be effective ifD =

∑i dixi with di ≥ 0. We will write simply D ≥ 0 if D is effective.

For every f ∈ F , we have a divisor div(f) =∑

x νx(f)x where νx(f) is the vanishing orpole order of f at the point x. If ux is a parameter of X at x then f ∼ uνxx up to themultiplication by invertible function at x. We have div(f) ∈ Div0(X).

For every Zariski open subset U ⊂ X, we also the group Div(U) and the notion of efectivedivisors of U . For every meromorphic function f ∈ F , we also have a divisor divU(f) =∑

x∈U νx(f)x.We will denote OX the structural sheaf of the the algebraic curve X whose generic fiber

is F . For every Zariski open subset U ⊂ X, the sections of OX(U) are regular algebraicfunctions on U . In other words

Γ(U,OX) = f ∈ F | divU(f) ≥ 0.For every x, the local ring of germs of regular functions at x is

OX,x = f ∈ F | νx(f) ≥ 0.This is a regular local ring of dimension one i.e. its maximal ideal is generated by oneelement. A generator of the maximal ideal of OX,x is called a parameter of X at x.

More generally, for every D ∈ Div(X), the sheaf OX(D) is defined by

Γ(U,OX(D)) = f ∈ F | D|U + divU(f) ≥ 0.For every x, the group of germs of its sections regular at x is

OX(D)x = f ∈ F | dx + νx(f) ≥ 0.Its is clear that for every U ⊂ X, H0(U,OX(D)) is a H0(U,OX)-module and for every x ∈ X,OX(D)x is a OX,x-free module of rank one. This means that for every D ∈ Div(X), OX(D)is a locally free OX-module of rank one, in other words a line bundle over X.

The following statement is a consequence of the Riemann-Roch theorem and the dualityof Serre that we will recall later.

Proposition A.1.1. For every D ∈ Div(X) the vector space

Γ(X,OX(D)) = f ∈ F | D + div(f) ≥ 0is a finite dimensional. If deg(D) < 0, we have Γ(X,OX(D)) = 0. If deg(D) > 2g−2 whereg is the genus of X, we have

dim Γ(X,OX(D)) = 1− g + deg(D).78

A.2. Line bundles. This theorem generalizes to any line bundle because every line bundleL is of the form OX(D). More precisely, let a ∈ L⊗OX F be a meromorphic section of L. For

every x ∈ X, let νx(a) be the integer so that we have the equivalence relation a ∼ lxuνx(a)x up

to multiplication by an invertible unction at x. Here lx is a generator of Lx as OX,x-moduleand ux is a generator of the maximal ideal of Ox. Let define D = div(a) =

∑x∈X νx(a)x.

Then we have a canonical isomorphism L = OX(D). Over the generic fiber, this is theisomorphism between one-dimensional F -vector spaces F → L ⊗OX F assigning 1 7→ a.Even if any line bundle L on X is of the form L = OX(D) for some divisor D, line bundle isnot naturally equipped with a meromorphic section so the line bundles and divisors are notequivalent notions. Nevertheless the integer deg(div(a)) does not depend on the choice of themeromorphic section a of L since two different meromorphic section differ by a meromorphicfunction. Therefore deg(L) = deg(div(a)) is well defined.

Theorem A.2.1 (Riemann-Roch). The cohomology groups Hi(X,L) are finite dimensionaland vanish if i /∈ 0, 1. We have

dim H0(X,L)− dim H1(X,L) = 1− g + deg(L).

The sheaf of 1-forms ΩX/k of X over k is a line bundle over X of degree 2g − 2. Forevery affine open subset U ⊂ X with ring of regular functions A = Γ(U,OX), we haveΓ(U,ΩX/k) = ΩA/k where ΩA/k is the A-module of Kahler differentials.

Theorem A.2.2 (Serre). There is a canonical non degenerate pairing between H0(X,L) andH1(X,L−1 ⊗ ΩX/k). In particular, we have a canonical isomorphism H1(X,ΩX/k)→ C.

A.3. Covering. Let us now review the Hurwitz theorem. Let f : Y → X be a finitemorphism between smooth projective curves over C. Pick a point y ∈ Y with image x ∈ X,and let ux be a local parameter of X at x and uy a local parameter of Y at y. We say thatf is etale at y if ux is a local parameter of Y at y, in other words ux and uy differ by aninvertible function at y. Since our base fields in C, this happens for all but finitely manypoints y ∈ Y . A non etale point y ∈ Y is also called a ramified point. There exists an integerey, the ramification index, such that ux ∼ u

eyy up to an invertible function.

Theorem A.3.1 (Hurwitz). Let f : X → Y be a finite morphism of degree d between smoothprojective curves over C. We have the relation

2gY − 2 = d(2gX − 2) +∑y

(ey − 1)

where we sum over all ramification points of Y .

One can pull back a 1-form from X to Y . This defines homomorphism f ∗ΩX → ΩY whichwhich is an injective map whose cokernel is a torsion sheaf ΩY/X supported by the ramifiedpoints. We have deg(f ∗ΩX) = d(2gX − 2) and deg(ΩY ) = 2gY − 2. It is not difficult toevaluate the length of the cokernel. Let y ∈ Y be a ramified point. We derives from therelation ux ∼ u

eyy that dux ∼ u

ey−1y duy which means that the length of the direct factor of

ΩY/X supported by y is ey − 1. The Hurwitz theorem follows.

A.4. Adelic desctiption. Following Weil, any line bundle admit adelic description as fol-lows. Recall that Fx is the completion of F with respect to the topology defined by themaximal ideal of the local ring OX,x and Ox its ring of integers. In constrast of the local

79

ring OX,x remembers about the curve X because its field of fractions is F , the completedlocal ring Ox is always isomorphic to the ring of formal series of one variable. Let AF denotethe ring of tuples (fx)x∈X with fx ∈ Fx for all x ∈ X and fx ∈ Ox for all but finitely manyx ∈ X. In particular LA contains both F and

∏xOx.

We can attach to line bundle L on X the following adelic data. First we have L = L⊗OX Fis a one-dimensional F -vector space. At each point x ∈ X, the one-dimensional vector spaceLx = L⊗F Fx is equipped with a Ox-submodule Lx = L⊗OX Ox which is a free Ox-moduleof rank one. The adelic data (L, (Lx)x∈X) is required to satisfy the following condition : forevery nonzero element a ∈ L, a is generator of Lx for all but finitely many x ∈ X. Wedenote LA = L⊗F AF that contains both L and

∏x Lx. We observe that the adelization of

line bundle has an obvious generalization to vector bundles.

Proposition A.4.1. The cohomology groups are given by the following formula

H0(X,L) = L ∩∏x

Lx(A.4.1)

H1(X,L) = LA/(L+∏x

Lx).(A.4.2)

For each x ∈ X, we have a canonical map ΩX ⊗OX Fx → C given by the residue at x. Bytaking the sum of residue, we have a map LA → C whose restriction to

∏x ΩX⊗OXOx vanish

by construction, and whose restriction to ΩX,x⊗OX F vanishes by the residue theorem. Thisdefines a canonical map H1(X,L)→ C that appears in Serre’s theorem.

Appendix B. Fourier transform and the Poisson summation formula

See [9, chapter 5] for more details and proofs.

B.1. Schwartz functions and the Fourier transform. A Schwartz function on R isa smooth (indefinitely differentiable) functions f : R → C so that f along with all itsderivatives f ′, f (2), . . . are rapidly decreasing, in the sense that

(B.1.1) supx∈R|x|k|f (`)(x)| <∞ for every k, ` ≥ 0

We denote S(R) the space of all Schwartz functions. Smooth functions with compact supportare also Schwartz functions.

Simple example of Schwartz functions are P (x)e−x2

where P (x) is a polynomial func-tion. The main property of the Schwartz class of function is its stability under the Fouriertransform f 7→ f with

f(y) =

∫ ∞−∞

e−2πixyf(x)dx.

Theorem B.1.1. If f ∈ S(R) then f ∈ S(R).

Proof. It can be easily checked that the Fourier transform of a Schwartz function is bounded.The theorem derives from the fact that the Fourier transform exchanges diffentation andmultiplication

yk(d

dy

)`f(y)

80

is the Fourier transform of1

(2πi)k

(d

dx

)k(−2πix)`f(x)

which is also a Schwartz function.

B.2. The Poisson summation formula.

Theorem B.2.1. Let f ∈ S(R) be a Schwartz function. The series

∞∑n=−∞

f(n) and∞∑

n=−∞

f(n).

are absolutely convergent. Moreover, we have the equality∞∑

n=−∞

f(n) =∞∑

n=−∞

f(n).

Proof. [9, p.154].

It is sometimes useful to relax the rapidly decreasing condition. A function f is said tobe a sufficiently decreasing condition if there exist constant ε > 0 and A > 0 such that

|f(x)| < A

1 + |x|1+ε.

The series∑n=∞

n=−∞ f(n) is absolutely convergent as long as the function f is sufficientlydecreasing. In fact the Poisson summation formula and the proof [9, p.154] holds if both f

and f are sufficiently decreasing.

Proposition B.2.2. For any function f that is twice continously derivable and if f, f ′, f (2)

have moderate decrease, the Poisson summation formula holds.

Appendix C. Review on Haar measures

C.1. Locally compact groups. Let G be a locally compact topological group. We denotethe action of G on itself by translation on the left by the formula l(g1)g = g1g. The action bytranslation on the right is given by r(g1)g = gg−1

1 . These actions induces two actions of G onthe space Cc(G) of continuous functions with compact support, given by l(g1)f(g) = f(g−1

1 g)and r(g1)f(g) = f(gg1). By duality, we have action on the left and on the right of G on thespace of linear functionals of Cc(G).

Proposition C.1.1 (Haar). There exists one and up to a factor only one left-invariantmeasure dl(G)g and we call it a left-invariant Haar measure.

Obviously, there is also a right invariant measure dr(G) well defined up to a positive factor.Left and right invariant measures are related by the Haar modulus. The right translationr(g1)dl(G) is still a left-invariant measure, so that there is a unique positive constant δG(g1)such that r(g1)dl(G)g = δG(g1)dl(G)g, and the map g1 7→ δG(g1) defines a character G→ R+.

Lemma C.1.2. If dl(G)g is a left invariant measure then δG(g)dl(G)g is a right invariantmeasure.

81

The group G is said to be unimodular if the Haar modulus is trivial. This is the casefor instant if G is its owned derived group [G,G] so that all characters G → R+ ought tobe trivial. It is also the case if G is compact because any there is no non trivial compactsubgroup in R+. In this case, we will write dg for both dl(G)g and dr(G)g.

In the case of Lie group i.e. a topological group equipped with a manifold structure. Heremanifold can be defined over F = R,C or a p-adic field. Let g be the tangent space of

G at the origin. Given by a non zero linear form∧dim(G) g → F where F = R,C or Qp

respectively, we have a left (resp. right) invariant volume form ωl(G) (resp. ωr(G)) by left(resp. right) translation. These forms are related by

ωr(G) = det(ad(g))ωl(G)

where ad(g) denote the adjoint action. Recall that ad(g) is the derivation of the conjugationaction x 7→ gxg−1 of G on itself.

We also have densities |ω|l(G) and |ω|r(G) associated to these volume forms. They are ofcourse related by

ωr(G) = | det(ad(g))|ωl(G)

and also the left (right.) invariant measure dl(G)g (resp. dr(G)g).

C.2. Quotient space. Let H be a closed subgroup of a locally compact group G. For allcharacters χ : H → C×, we consider the space Cc(H\G,χ) consisting of locally constantfunction f : G→ C that satisfy the equation f(hg) = χ(h)f(g) for all h ∈ H and g ∈ G andsuch that there exists a compact Kf so that f is supported on HKf .

Proposition C.2.1. (1) We have a linear surjective map

(C.2.1) Λ : Cc(G)→ Cc(H\G, δH)

that assigns to every f ∈ Cc(G) the function (Λf)(x) =∫Hf(hg)dl(H)h.

(2) This application satisfies the property

Λ(l(h)f) = Λ(f)

and for every continuous linear map Cc(G)→ E invariant under l(H) must factorizethrough Λ. In particular the left invariant measure as a linear functional Cc(G)→ Cfactorizes through Cc(H\G, δH).

(3) If G is unimodular, the induced linear map Cc(H\G, δH)→ C is invariant under theright action of G on H\G. All such G-invariant linear forms are proportional.

Proof. The main observation here is that Λ does not commute with the left translation byh1 ∈ H if δH is non trivial. Let us check first the invariant property Λ(l(h−1

1 )f) = Λ(f) forall h1 ∈ H. For all g ∈ G, we have

Λ(l(h−11 )f)(g) =

∫H

f(h1hg)dl(H)h

=

∫H

f(h′g)dl(H)h′

= Λ(f)(g).82

For every h1 ∈ H, we have

(l(h−11 )(Λf))(g) = (Λf)(h1g)

=

∫H

f(hh1g)dl(H)h

= δH(h1)

∫H

f(h′g)dl(H)h′

= δH(h1)Λf(g)

where we used the change of variables h′ = hh1 and the transformation rule of measuresδH(h1)dl(H)h

′ = dl(H)h.Let f ∈ Cc(G) be a function supported by a compact set Kf . The function Λf is then

supported by HKf . Thus Λ defines a linear map Cc(G)→ Cc(H\G, δH).For every x ∈ H\G, there exists an open subset Ux so that the quotient map G → H\G

admits a section over ux : Ux → G. We can identify the preimage of Ux in G with H×ux(Ux).We can also assume Ux is contained in a compact subset of H\G. Let φ ∈ Cc(H\G, δ−1

H ) sothat f is supported by HKf where Kf is a compact subet of G. Since the image of Kf inH\G is compact, there exists a finite number of x ∈ H\G so that the union of Ux covers theimage of Kf . Using the partition of unity, we an assume that φ is supported on H × εx(Ux).We can now choose a function f in the form f = fH ⊗ φ so that fH is continuous andcompactly supported and such that

∫HfH(h)dl(H)h = 1. Since Ux is contained in a compact,

f is compactly supported in G and we have Λf = φ.We can prove that for every continuous linear map Cc(G) → E that is invariant under

l(H) has to factorise through Λ : Cc(G) → Cc(H\G, δH). In particular the left invariantmeasure Cc(G)→ C factorises through a canonical linear map I : Cc(H\G, δH)→ C.

Assume that G is unimodular i.e. dl(G)g = dr(G)g. The straightforward calculation

I(r(g1)Λ(f)) =

∫G

f(gg1)dg

=

∫G

f(g′)dg′

= L(Λ(f)).

shows that the map I : Cc(H\G, δH)→ C is invariant under the right action of G

We are mainly interested in the case where there exists a compact subgroup K of G sothat G = HK. In that case the support condition is automatically satisfied so that wehave Cc(H\G, δH) = C(H\G, δH). Moreover, the linear form L : C(H\G, δH)→ C can alsodefined more explicitly as integration over K.

Proposition C.2.2. Suppose that there exists a compact subgroup K so that G = HK.Then the restriction to K defines an isomorphism

C(H\G, δH)→ C(H ∩K\K).

The linear form I : C(H\G, δH)→ C and the integration on K with respect to its invariantmeasure are proportional with respect to this identification.

Proof. Let f ∈ C(H\G, δH). It derives from G = HK and f(hk) = δH(h)f(k) that f isuniquely determined by its restriction to K. Since K is compact, its closed subgroup H ∩K

83

is also compact. It follows that the restriction of the character δH to H ∩K is trivial. Thusthe restriction to K defines a linear bijection C(H\G, δH)→ C(H ∩K\K).

The linear form I : C(H\G, δH) → C defines a linear form C(H ∩ K\K) that is K-invariant on the right. It must be proportional with the linear form C(H ∩K\K) definedby integration over K.

Appendix D. Operators

For more details, see [8, chapter VI].

D.1. Compact operators.

Definition D.1.1. Let X, Y be Banach spaces. A bounded operator T : X → Y is calledcompact if it takes bounded subset in X into a precompact set in Y i.e. a subset of Y whoseclosure is compact. In other words, for every bounded sequences xn ⊂ X, Txn has aconvergent subsequence in Y .

Important examples of compact operator are integral operator on a compact Haussdorffspace, in particular on the unit interval [0, 1]. Let K(x, y) be a continuous function of twovariables x, y ∈ [0, 1]. The operator

(Kφ)(x) =

∫ 1

0

K(x, y)φ(y)dy

is a bounded operator on the Banach space C[0, 1] with the uniform norm

||φ||∞ = maxx∈[0,1]

φ(x).

We have indeed

||Kφ||∞ ≤ maxx,y∈[0,1]

K(x, y)||φ||∞.

Let B1 be the set of φ ∈ C∞[x, y] with norm ||φ||∞ ≤ 1. The operator K takes B1 into anequicontinuous family. By compacity, for every ε > 0, there exists δ > 0 so that |K(x, y)−K(x′, y)| < ε for every y ∈ [0, 1] as long as |x−x′| < δ. This implies |(Kφ)(x)−(Kφ)(x′)| ≤ εfor every φ ∈ B1. By the Ascoli theorem, a bounded equicontinuous family of functions on[0, 1] is precompact. In other words, K is a compact operator on [0, 1].

Lemma D.1.2. Let S, T ∈ L(X). IF S or T is compact then ST is compact. In otherwords, compact operators form a left and right ideal of L(X).

Proposition D.1.3. Operators of finite rank are compact and a norm limit of operators offinite rank is also compact. Inversely, every compact operator on a separable Hilbert space isthe norm limit of a sequence of operators of finite rank.

Proof. Balls in finite dimensional vector spaces are compact, and therefore operators of finiterank are compact.

Let T : E → F be norm limit of compact operators Tn : E → F . Let xm be a boundedsequence in E. After the extracion of a subsequence, we can assume that for every integersm1,m2 ≥ n, we have |Txm1 −Txm2 | ≤ 1/n. It follows that Txm is a Cauchy sequence whichought be convergent as F is complete.

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Let xi∞i=1 be an orthonormal basis of H. It suffices to prove that the finite rank operators

v 7→n∑i=1

〈xi, v〉Txi

converge to T with respect to the norm topology. It suffices to prove that the norm of therestriction of T to the Hilbert subspace generated by xn+1, xn+2, . . . tends to zero as n goesto infinity. This derives from the compacity of T .

Proposition D.1.4. Let M be a locally compact topological space equipped with a measure,and H = L2(M). For every K ∈ L2(M ×M), the operator φ 7→ AK(φ) with

AK(φ)(x) =

∫M

K(x, y)φ(y)dy

is a compact operator.

Proof. It follows from the Cauchy-Swartz inequality that ||AK || ≤ ||K||2. Thus the operatorsAK is continuous.

Let φi∞i=1 be an orthonormal basis for L2(M). Then φi(x)φj(y) is an orthonormalbasis for L2(M ×M) so

K =∞∑

i,j=1

αijφi(x)φj(y).

Consider the kernels

Kn =n∑

i,j=1

αijφi(x)φj(y)

whose associated operators AKn are of finite rank. Since Kn → K as n → ∞ in L2-norm,AKn → AK in norm topology as n→∞. It follows that AK is a compact operator.

Theorem D.1.5 (The Fredholm alternative). If A is a compact operator on H, then either(1− A)−1 exists as bounded operator or Av = v has a nonzero solution.

Proof. Let F be a finite rank operator such that ||A−F || < 1. Then the operator 1−(A−F )is invertible and its inverse is the infinite series 1 +

∑∞n=1(A− F )n. We have

(1− A)(F (1 +∞∑n=1

(A− F )n)) = (1− (A− F )− F )(1 +∞∑n=1

(A− F )n)

= 1− F (1 +∞∑n=1

(A− F )n)

where F (1 +∑∞

n=1(A − F )n) is a finite rank operator. It is thus enough to consider thecase of finite rank operator instead of compact operators. In this case (1 − A) is invertibleif and only if the determinant of the restriction of 1 − A to the range of A, which is finitedimensional, is non zero. This is equivalent to the existence of a nonzero solution of theequation Av = v.

Theorem D.1.6 (Analytic Fredholm theorem). Let D be an open connected subset of C.Let A : D → L(H) be an analytic family of compact operators. Assume that there existsz ∈ D so that (1 − A(z))−1 exists as bounded operator. Then (1 − A(z))−1 is meromorphic

85

in D, analytic in the complement D − S of a discrete set S ⊂ D. Its residue at every pointz ∈ S is an operator of finite rank. Moreover, at every point z ∈ S, there exists a nonzerosolution of the equation A(z)v = v.

Proof. Similar to the aproof of the Fredholm alternative, see [8, p.202].

Theorem D.1.7 (Riesz-Schauder). Let A be a compact operator on a Hilbert space H. Thereexists a bounded subset σ(A) ⊂ C with no accumulation points but 0 so that (λ−A)−1 existson C− σ(A). Moreover, every nonzero λ ∈ σ(A) is an eigenvalue of finite multiplicity.

Appendix E. Abelian L-functions

E.1. Characters and Hecke characters. Let F be a global field. We denote by Fv itslocal fields and AF its ring of adeles. We seek a description of characters χ : F×v → C aswell as characters of the ideles class group F×\A× → C. Our convention is that charactersare continuous homomorphisms with values in C×, and unitary characters are continuoushomomorphisms with values in the unit circle C1.

Recall that the local field Fv is equipped with an absolute value x 7→ |x| ∈ R+. Its kernelUv is a compact subgroup of Fv. If Fv = R, then Uv = ±1. If Fv = C then Uv = C1. IfFv in non archimedean, then Uv = O×v . There exists a homomorphism F×v → Uv noted byx 7→ uv(x) constructed as follows. If v is archimedean, uv(x) is the unique element of Uvsuch that x = uv(x)y with y ∈ R+. In the non archimedean case, we need to make a furtherchoice of a prime element εv ∈ F×v . Then uv(x) is the unique element of Uv such that

x = uv(x)εordv(x)v .

Thus, Fv = Uv × R+ if v is archimedean and Fv = Uv × Z if v is archimedean.A character χ : Fv → C× trivial on Uv is called unramified. All unramified character is of

the formx 7→ |x|s = es log |x|

for a certain complex number s. This complex number is well defined of Fv is archimedean.In the non-archimedean case, it is only well defined modulo 2πi log |εv|. The characterx 7→ |x|s is unitary if and only if s is purely imaginary. In the archimedean case, the spaceof unramified characters is the complex plane C with the imaginary line iR as the space ofunitary unramified characters. In the non archimedean, the space of unramified charactersis the cylinder C/2πi log |εv| with the imaginary circle iR/2πi log |εv| as the space of unitaryunramified characters.

The characters of the compact group Uv is automatically unitary. They form the unitarydual Uv of Uv; Uv is a discrete group as Uv is a compact group. If Fv = R, Uv = ±1, its

dual is also Uv = ±1. If Fv = C, Uv = C1, its dual is Uv = Z. If Fv is non archimedean,Uv is an inverse limit of finite group, its dual is a direct limit of finite groups.

Proposition E.1.1. The characters χ : Fv → C× are all of the form χ(x) = χu(uv(x))|x|swhere χu is a character of the compact group Uv and where |x|s is an unramified character,the complex number s is completely determined by χ in the archimedean case but only modulo2πi log |εv| in the non archimedean case. It is unitary if and only if s is purely imaginary.

Proof. The statement follows from the decomposition Fv = Uv ×R+ if v is archimedean andFv = Uv × Z if v is archimedean.

86

A Hecke character is a character χ : F×\A×F → C×. It induces a character χv : F×v → C×for each place v so that we have χ = ⊗vχv. Since χ is continuous, χv is unramified for allbut finitely many v. A Hecke character χ is called unramified if all its local components χvare unramified.

If F is the field of rational functions of a curve C as in 5.2, an unramified Hecke characteris a character χ : PicC → C×. With a choice of a line bundle of degree one, we have asplitting PicC = Pic0

C × Z. In this case χ will also split as χ = χ0 ⊗ |.|s where χ0 is acharacter of the finite group Pic0

C and where |.|s is the character x 7→ |x|s = q− deg(x)s.

E.2. Eigendistributions for local fields. In [3], Kudla gave an exposition of Tate’s thesisfollowing the treatment to Weil in his Bourbaki talk.

Let S(Fv) be the space of Schwartz-Bruhat functions on the local field Fv; its dual S(Fv)′

is the space of tempered distributions. If v is non archimedean, S(Fv) is the space C∞c (Fv)of complexed valued locally constant functions with compact support. The tempered dis-tributions are linear functionals on it. If v is archimedean, S(Fv) is the space of complexedvalued smooth functions which, together with all its derivatives, are rapidly decreasing. Thisis a Frechet space. The tempered distributions are the continuous linear functionals on it.

The group F×v acts on Fv and induces an action on the space of Schwartz-Bruhat functionsS(Fv) as well as the space of tempered distributions S(Fv)

′. For every character χ : F×v →C×, let us denote S(Fv)

′(χ) the space of eigenvectors corresponding to the eigenvalue χ.

Proposition E.2.1. Assume v non archimedean. For any character χ : F×v → C×, we havedimS(Fv)

′(χ) = 1.

Proof. We have an inclusion C∞c (F×v ) → S(Fv) from the space of function on F× withcompact support into the space of Schwartz-Bruhat functions on Fv. By duality, there is anexact sequences

0→ S(Fv)′0 → S(Fv)

′ → C∞c (Fv)′ → 0

where S(Fv)′0 is the space of distributions on Fv supported by 0. Since v is non-archimedean,

S(Fv)′0 is one-dimensional and generated by the δ-distribution δ0.

For evert character χ : Fv → C×, this induces a left exact sequences of χ-eigenspaces

0→ S(Fv)′0(χ)→ S(Fv)

′(χ)→ C∞c (Fv)′(χ).

We have dimS(Fv)′0(χ0) = 1 for the trivial character χ0, and dimS(Fv)

′0(χ) = 0 for nontrivial

characters χ 6= χ0. The space C∞c (Fv)′(χ) is one-dimensional. If d×x is a Haar measure

on F×v , this space is generated by the distribution χ(x)d×x. We have the inequality 1 ≤dimS(Fv)

′(χ0) ≤ 2 and in the second case we have dimS(Fv)′(χ) ≤ 1. For nontrivial

character, it remains to prove that dimS(Fv)′(χ) ≥ 1 by constructing explicitly a nonzero

element of its. For this, we will consider two different cases : χ is unramified and χ isramified. Later on, we will consider separately the case of the trivial character.

The linear application S(Fv) → C∞c (F×v ) mapping f on the function f(x) − f(ε−1v x) is

F×v -equivariant. By composing with the application C∞c (F×v )→ C given by the integrationagainst the measure χ(x)d×x, we get an element S(Fv)

′(χ) for any character χ : F× → C×.This element may or may not be zero. Let’s evaluate it on the function f = 1Ov . The functionx 7→ f(x)− f(ε−1

v x) is the characteristic function 1O×v of the maximal compact subgroup O×vof F×v . If χ is unramified, the integration of 1O×v against χ(x)d×x is one. We defined in this

87

case a nonzero element of dimS(Fv)′(χ) that we will denote by z0(χ) ∈ S(Fv)

′(χ). It followsthat if χ in unramified and nontrivial dimS(Fv)

′(χ) = 1.Let us consider now the case of a ramified character. In this case, for every f ∈ S(Fv),

the integral ∫F×v −εnvOv

f(x)χ(x)d×x

is convergent and independent of n for large n since f is constant on εnvOv for n large enough.This defines a nonzero element z0(χ) ∈ S(Fv)

′(χ) for every ramified character χ.Finally, we consider the case of the trivial character χ0 ...

By will write z0(s, χ) = z0(χ|.|s) to emphasize on the dependance on the twisting by |.|swhen the complex number s varies. For an unitary character χ of F×v , the local zeta integral

(E.2.1) z(s, χ; f) :=

∫F×v

f(x)χ(x)|x|sd×x

is absolutely convergent for all f ∈ S(Fv) provided <(s) > 0. In this range, the distributionf 7→ z(s, χ; f) defines a nonzero element z(s, χ) ∈ S(Fv)

′(χ|.|s). A direct calculation showsthat

(E.2.2) z0(s, χ) = Lv(s, χ)−1z(s, χ)

when the right hand side is defined i.e. for all <(s) > 0. This expression implies an analyticcontinuation of z(s, χ) as L(s.χ)z0(s, χ).

Proposition E.2.2. Assume v archimedean. For any character χ : F×v → C×, we havedimS(Fv)

′(χ) = 1.

Proof. As in the non archimedean case, we have an inclusion C∞c (F×v ) → S(Fv) from thespace of function on F× with compact support into the space of Schwartz-Bruhat functionson Fv. By duality, there is an exact sequences

0→ S(Fv)′0 → S(Fv)

′ → C∞c (Fv)′ → 0

where S(Fv)′0 is the space of distributions on Fv supported by 0. In contrast with the non

archimedean case, the space of distributions supported by zero S(Fv)′0 is no longer one-

dimensional. If v is real, S(Fv)′0 is generated by the derivatives Dkδ0 of the δ-function δ0. If

v is complex, S(Fv)′0 is generated by DkDlδ0.

The above sequence induces a left exact sequences of χ-eigenspaces

0→ S(Fv)′0(χ)→ S(Fv)

′(χ)→ C∞c (Fv)′(χ).

It is easy to see that the space C∞c (Fv)′(χ) is one-dimensional. If d×x is a Haar measure on

F×v , this space is generated by the distribution χ(x)d×x. If v is real, S(Fv)′0(χ) 6= 0 only for χ

is one of the character χk(x) = x−k for positive integers k for which S(Fv)′0(χk) is generated

by Dkδ0. We have similar conclusion if v is complex, with the characters χk,l(x) = x−kx−l.As in the non archimedean case, we consider the zeta integral

(E.2.3) z(s, χ; f) :=

∫F×v

f(x)χ(x)|x|sd×x

that is convergent assuming χ unitary and <(s) > 0. In contrast with the non archimedeancase, it is the whole Taylor series of f that accounts for the poles of z(s, χ; f) rather thanjust the value f(0). The proposition follows from the following two lemmas.

88

In the real case, a unitary character χ : R× → C1 must be either the trivial character orthe sign character. In both case, we put L(s, χ) = π−

s2 Γ( s

2). In the complex case, a unitary

character χ : C→ C1 must be of the form χ(z) = znz−n.

Lemma E.2.3. The distribution z0(s, χ) = L(s, χ)−1z(s, χ) has an entire analytic continu-ation to the whole s-plane, and for all s, defines a basis vector for the space S(Fv)

′.

Proof. The proof is based from an elementary calculation contained in the following lemma.

Lemma E.2.4. (See [2, Lemma 3.1.7]) Suppose that f is a continuous function on [0, 1]which has a uniform convergent Taylor expansion

f(x) =∞∑k=0

akxk.

Then ∫ 1

0

f(x)xs−1dx

has meromorphic continuation to all s, with poles only at the values s = −k. The residue atthe pole s = −k is the coefficient ak.

Fix a nontrivial additive unitary character ψ : Fv → C1. This choice induces an isomor-phism of Fv with its dual Fv = Hom(Fv,C1); we associate to an element y ∈ Fv the unitarycharacter x 7→ ψ(xy). The Fourier transform

f(y) =

∫Fv

f(x)ψ(xy)dx

of a function f ∈ S(Fv) is well defined and again lies in S(Fv). There is a unique choice

of Haar measure dx that is self-dual with respect to this Fourier transform so thatˆf(x) =

f(−x). The Fourier transform on S(Fv) gives rise to the Fourier transform on the dual spaceS(Fv)

′ of tempered distributions.

Proposition E.2.5. If λ is an χ-eigendistribution, its Fourier transform λ is an χ−1χ1-eigendistribution where χ1 is the absolute value character : χ1(x) = |x|.

Corollary E.2.6. There exists a nonzero constant ε(s, χ, ψ) such that

z0(1− s, χ) = ε(s, χ, ψ)z0(s, χ).

This constant, called the local epsilon factor, depends on χ, ψ and holomorphically on s.

For a given function f ∈ S(Fv), the local functional equation can be written as the relation

z(1− s, χ−1, f) = γ(s, χ, ψ)z(s, χ, f)

where the gamma factor is defined as

γ(s, χ, ψ) = ε(s, χ, ψ)L(1− s, χ−1)

L(s, χ).

In contrast with the ε-factor, the γ-factor may have pole or zero.89

E.3. The global functional equation and the Poisson summation formula. Let Fbe a global field, AF its ring of adeles. The space S(A) of Schwartz-Bruhat functions on Acontains all functions of the form f = ⊗vfv where fv ∈ S(Fv) for all places v and fv = 1Ovfor almost all v. The finite linear such factorizable functions are dense on S(A). Again thetopological dual S(A)′ consisting on continuous linear functionals on S(A) is the space oftempered distributions. Let λv be a distribution on S(Fv) such that 〈λv, 1Ov〉 = 1 for almostall v, then there exists a tempered distribution λ = ⊗λv such that 〈λ, f〉 =

∏v〈λv, fv〉.

Proposition E.3.1. For all character χ of A×F , the space of global χ-eigendistributionsS(AF )′(χ) has dimensional one. For any s ∈ C, S(AF )′(χ|.|s) is generated by the standarddistribution

z0(s, χ) = ⊗vz0(s, χv).

For a character χ : F×\A×F and a function f ∈ S(A), we can define the global zeta integralby

(E.3.1) z(s, χ, f) =

∫A×f(x)χ(x)|x|sd×x.

If f is factorizable f = ⊗vfv, this is a product of the local zeta integrals z(s, χv, fv). Sincez0(s, χv, fv) = 1 as long as χv is unramified and fv = 1O×v which happens for v /∈ S whereS is a finite set of places, the convergence of the infinite product z(s, χ, f) boils downto the convergence of the local zeta integrals and the convergence of the partial L-functionLS(s, χ) =

∏v/∈S Lv(s, χv). We know that this product is absolutely convergent for <(s) > 1.

On this half-plane of convergence, we have

z(s, χ) = Λ(s, χ)z0(s, χ)

where Λ(s, χ) =∏

v Lv(s, χv) is the complete L-function.We fix a global additive character ψ : k\A→ C1. By restricting to local components, we

can write ψ = ⊗vψv where ψv is unitary character of Fv. This choice permits us to identify Awith its dual A = Hom(A,C1) compatibly with the local identifications Fv = Hom(Fv,C1).The global Fourier transform S(A)→ S(A) is compatible with the local ones i.e., if f = ⊗vfvthen f = ⊗vfv. It induces a Fourier transform of tempered distributions S(A)′ → S(A)′.

Theorem E.3.2. The distribution z(s, χ) defined by the global zeta integral (E.3.1) for<(s) > 1 has a meromorphic continuation to the whole s-plane and satisfies the functionalequation

z(1− s, χ−1) = z(s, χ).

Proof. Since Schwartz functions have rapid decay at ∞, the integral

z>1(s, χ, f) =

∫A×,|x|>1

f(x)χ(x)|x|sd×x

is convergent for all s.

Recall that E.2.6 has the form z0(1 − s, χ−1v ) = εv(s, χv, ψv) = z0(s, χv) where the local

epsilon factors εv(s, χv, ψv) are equal to one for almost all v. We define the global epsilonfactor by ε(s, χ) =

∏v εv(s, χv, ψv) and obtain the functional equation

z0(1− s, χ−1) = ε(s, χ)z0(s, χ).90

By comparing this expression with E.3.2, we get the functional equation of the completeL-function.

Corollary E.3.3.Λ(s, χ) = ε(s, χ)Λ(1− s, χ−1).

References

[1] Borel, Automorphic forms on SL2(R)[2] Bump, Automorphic forms and representations.[3] Kudla, Tate’s thesis[4] Lang, SL2(R)[5] Miyake, Modular forms[6] Serre, Cours d’arithmetique[7] Shimura, Introduction to the arithmetic theory of automorphic forms[8] Reed, Simon, Functional analysis[9] Stein, Shakarchi, Fourier analysis: an introduction[10] Stein, Shakarchi, Complex analysis[11] Wallach, Real reductive groups I.[12] Warner, Harmonic analysis on semisimple Lie groups I.

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