Maass Cusp Forms with Integer Coefficients

By a Maass cusp form we will mean a weight zero cuspidal eigenform ofthe Laplacian for a congruence subgroup F0(N) of SL(2, Z), possibly witha primitive central character x. We assume that is a new form and that itis also a Hecke eigenform. Corresponding to is an automorphic cuspidalrepresentation rr of GL2(AQ). These ‘s are quite mysterious, even their existence being a subtle issue (Phillips & Sarnak 1985, Luo 2001). Cusp forms ingeneral are the building blocks of modem automorphic form theory and theseMaass forms in particular are especially important in the analytic applicationsof the theory (Iwaniec 1995, Iwaniec & Sarnak 2000). Unlike their holomorphic weight k> 1 counterparts, very little is known about the algebraic natureof their coefficients A,k (n), ii 1. Here the coefficients are those in the L

series: L(s, 4,) = A(n)n5 or equivalently the eigenvalues of the Hecke

operators T(n).

Some remarkable Maass forms with integer coefficients are known. Theyare related to finite subgroups of GL2(C). The finite subgroups of PGL2(C)are well known (Klein 1913). Using the classification of these groups and theirinverse images in G4m)(C) := (g e GL2(C)J(detg)m = 1), m = 1, 2, oneconcludes the following:

Any finite subgroup of GL2(C) whose elements have integer determinant and trace is conjugate to one of the following maximal such sub-

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Peter Sarnak

U2=

(a)

{±[ ],±[ ],±[C G4’(C)

o i 1 ±t 0I 0]’ [—i

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The image of U2 and of V2 in P GL2(C) coincide and is the Klein FourGroup, D2:

U3 {±[ ],+[?0

±[ 1 ],[ 1 ],}V3

= {±[ ],±[i][-i

(b)

±[ ‘],±[101 —1

11’

Note U3 C GL1’(C), V3 C GL(C) and their imagedihedral group D3.

lit xi+ix2(c)

= 12 L —X3 + iX4

in PGL2(C) is the

X3+iX4 1 1Ix=±lUU2;

x1—1x2 ] J

U4 C G4’(C) and its image in PGL(2, C) is the tetrahedral group A4.Let GQ be the Galois group of Q/Q and let p : GQ —÷ GL2(C) be an

irreducible 2-dimensional Galois representation. The L-function L(s, p) hasinteger coefficients if and only if the image of p can be conjugated to lie in oneof the above finite groups. Langlands (1980) has shown that corresponding toany p as above (the tetrahedral case being the critical one) is an automorphiccuspidal representation IT of GL2(A) such that L(s, zr) = L(s, p). Moreoverit is known, see Casselman (1977), that such a rr corresponds to a Maass cuspfrom with Laplace eigenvalue equal to 1/4 if E = det p : GQ —. GL 1(C)is even (i.e. E(J) = 1, where UEGQ is a complex conjugation). If E is odd,then rr corresponds to a holomorphic cusp form of weight 1 (see Sene 1977).In particular, if the image of p is contained in £12, U3 or U4, then det p = 1and r corresponds to a Maass form. Put another way, if p is odd, then its im

Z2

IMaass Cusp forms with Integer Coefficients 123

age in PGL2(C) is either D2 or D3 and hence by Deligne & Serre (1994)any weight 1 holomorphic cusp form with integer coefficients must be dihe

The above provides us with a rich set of Maass cusp forms with integercoefficients. In what follows we reproduce the content of a letter to H. Kimand F. Shahidi (Samak 2001). It shows how their recent works (see Kim &Shahidi 2001a, 2001b; Kim 2000; Henniart 2001) on the 4- and 5-dimensionalfunctorial lifts sym3 and sym4 of GL2(C) = LG(G = CL2) allow one toclassify Maass cusp forms with integer coefficients.cide and is the Klein Four

Theorem Let be a Maass cusp form with integer coefficients. Then corresponds to an even irreducible 2-dimensional Galois representation whose image is contained in one of the groups described in (a), (b), or (c) above. In particular the Laplace eigenvalue A,(oo) is 1/4 and satisfies the Ramanujan—Selberg Conjectures.

Proof Let zr be the automorphic cuspidal representation of GL2(A) corresponding to and let x be its central character. We examine the cuspidality ofthe automorphic functorial lifts symk 7t to GLk+1 (A), k = 2, 3, 4. If sym2 sris not cuspidal, then according to Gelbart & Jacquet (197$) there is a quadraticcharacter i 1 of AQ*/Q* such that 7C 0 r,z 7r. In this case ij determinesa quadratic extension K of Q and rr a Hecke character A of A /K* such thatL(s, A) = L(s, ar), see Labesse & Langlands (1979). Since zr has integer coefficients it is easy to see that A must be of finite order. The 2-dimensionalGalois representation p = IfldK/Q E of GQ, where EA is the character of GKcorresponding to A via class field theory, satifies L(s, p) = L(s, A) = L(s, ir).This establishes the Theorem in this case. So if sym2 r is not cuspidal, we aredone. Now A(n) e R so that L(s, r x ) = L(s, sym2 ?T)L(s, x) where ñis the contragredient of 7T. The Rankin—Selberg L-function L(s, rr x ) has asimple pole at s 1. Hence, if x 1, L(s, sym2 7r) has a pole at $ = 1, butthen by Gelbart & Jacquet (1978) sym2 sr cannot be cuspidal and we proceedas above. Thus we may assume that x 1 and that sym2 7t is cuspidal. Next ifsym3 zr is not cuspidal on GL4(A), then as is shown in Kim & Shahidi (2001a),

7T corresponds to a representation p of the Weil group WQ of tetrahedral type.However since zr has integer coefficients we have that det p = ±1 (in fact sincex = 1 detp = 1). Hence p(WQ) C GL(C) and its projection in PGL2(C)is A4. Thus p must be a finite representation of WQ and hence a representation of C.

Thus 7T corresponds to a 2-dimensional Galois representation phaving integer trace and determinant and since it is of tetrahedral type, its image must be U4. To continue we can assume that both sym2 n and sym3 sr are

0 1±t 0 1i 0]’ t—1 0

0 lJ±t 0 11 0]’ [—i 0

dral.

—i i

image in PGL2(C) is the

= ±i} U U;

tetrahedral group A4.GQ — GL2(C) be an

[‘he L-function L(s, p) hasi be conjugated to lie in onehown that corresponding toical one) is an automorphics, Tr) = L (s, p). MoreoverDrresponds to a Maass cusp= detp : GQ —+ GL1(C)

conjugation). If e is odd,weight 1 (see Sene 1977).U3 or U4, then detp = 1

‘ay, if p is odd, then its im

124

cuspidal. If sym4 zr is not cuspidal then as shown in Kim & Shahidi (2001b),rr corresponds to a representation of WQ of octahedral type, that is its image inPGL(2, C) is S4. However, no lift of this group to GL(2, C) can have integerdeterminant and trace, and hence this case cannot happen for our zr. We are leftwith a ir whose central character x is trivial and such that symc rr, k = 2, 3,4are all cuspidal. We show that such a ir cannot have integer coefficients. Proceeding as done in Kim & Shahidi (2001b) we form the Ranldn—Selberg Lfunctions L(s, sym Zr x symc zr), 2 j, k 4, whose analytic propertiesincluding their nonvanishing on 1)1(s) = 1 are known (Shahidi 1990). From thefactorizations of these L-functions, see Kim & Shahidi (2001b), we deducethat L(s, symk rr) is analytic and nonvanishing on 1)1(s) = I for 1 k 8.Hence by standard analytic methods we have that for any polynomial T(x) ofdegree at most 8,

p_<N

T((?))logP__fT(x)dbc(x)

p prime

where di(x) = — is the ‘Sato—Tate’ measure.Set

2 2 2T(x)=x (x—1) (x+1)(4—x).

Note that T(m) < 0 for all in e Z while T(x) > 0 for x e [—2, 2]. Fromthe first we see that if A(p) e Z for all p, then the left-hand side of (1) isless than or equal to 0, while from the second we see that the right-hand sideof (1) is positive. This contradiction shows that such a Zr cannot have integralcoefficients.

Remarks

(1) S.D. Miller has pointed out to me that with a little more care one can use apolynomial of degree 6 rather than 8 in the previous argument. Instead of T(x)consider

P(x) = (4 — x2)x2(x2 — 1).

Again P(m) 0 for in e Z. While P(x) is no longer nonnegative on [—2, 2]a calculation shows that 1—2 P(x)d1u(x) = 1, which is still positive. So withP replacing T, the argument above shows that for a Maass cusp form Zr withinteger coefficients, L(s, symk Zr) must have a pole at s = 1 for some 2 k6. This reduces the range in k (which by the way allows one to carry out theanalysis above without appealing to the sym4 lift) to a sharp one. Indeed, if Zr

—

___fr

V..—

Peter Sarnak

(1)

--.

bass Cusp Forms with Integer Coefficients 125

corresponds to a tetrahedral Galois representation with image U4, then it has

integer coefficients and L(s, symk zr) has no pole for 1 k 5.

(2) If is a Maass cusp form as above and A, (oo) > , then according

to the Theorem, the A(n)’s cannot all be integers. We expect much more,

namely that has some transcendental coefficients. For the special case that

L(s, ) = L(s, x) for x a Grossencharacter, not of type A0, Well (1980), of a

real quadratic field K/Q, this is indeed true (these ‘s correspond to dihedral

representations of WQ with infinite image in GL2(C)). For example, let K =

Q(J), and E0 = 1 + be a fundamental unit in OK. For m a nonzero

integer, define xm by

irnyr

a logE0

xm((a))= —;a

for 0 $ a E OK (K has class number 1). Here a’ is the Galois conjugate of a.

The corresponding Maass form has Laplace eigenvalue

1 trrm\2

4 logo

Forp a rational prime which splits in K

irnyT im,r

ao a’io_7 + —

a a

where a E OK such that aa’ =p.

By the Gel’fond—Schneider theorem (.u log(—l) is transcendental\logEo log EQ

see Waldschmidt (2000), we see that A(oo) is transcendental. To see that

one of the A(p)’s is transcendental, let ai, a2, a3 > 0 be in OK with

N(a) = aa = p3 being distinct rational primes. Let x1 = log(a3/a)

for j 1, 2, 3 and Yi = 1, y = Clearly y’, Y2 are linearly independent

over Q and one checks that x, x2, X3 are too. Hence by the six exponential

theorem, see Waldschmidt (2000), one of the numbers

—j,—-

Kim & Shahidi (2001b),

al type, that is its image in

3L(2, C) can have integer

ppen for our rr. We are left

th that sym rr, k = 2, 3, 4

integer coefficients. Pro

m the Rankin—Selberg L

whose analytic properties

i (Shahidi 1990). From the

tahidi (2001b), we deduce

IR(s) = 1 for 1 k < 8.

or any polynomial T(x) of

[T(x)d(x) (1)

asure.

_x2).

> 0 for x e [—2,2]. From

the left-hand side of (1) is

see that the right-hand side

ch a r cannot have integral

ittle more care one can use a

ts argument. Instead of T (x)

nger nonnegative on [—2, 2]

iich is still positive. So with

r a Maass cusp form rr with

eats = 1 for some 2< k

i allows one to carry out the

to a sharp one. Indeed, if r

a1 a a3 (a1’\i

a’ a’ a’ k.a;)

(a2T (a3T

is transcendental. Hence one of)(p1), A(p2) or A(p3) is transcendental.

Bibliography

Casselman, W. (1977) Algebraic Number Fields, A. Frölich (ed.), AcademicPress, 663—704.

Deligne, P. and Serre, J-P., (1994) ‘Formes modulaires de poids 1’ Ann. Sci.E’colé Norm. Sup. 4 (7), 507—530.

Gelbart, S. and Jacquet, H. (197$) ‘A relation between automorphic forms onGL(2) and GL(3)’,Ann. Sci. Ecolé Norm. Sup. 11,471—552.

Henniart, G. (2001) ‘Progrës récents en fonctorialité de Langlands’ SeminarBourbald.

Iwaniec, H. (1995) Introduction to the Spectral Theory ofAutomorphic Forms,Mathematica Iberoamericana.

Iwaniec, H. and Samak, P. (2000) ‘Perspectives on the analytic theory of Lfunctions’, GAFA 11, 705—741.

Klein, F. (1913) Lectures on the Icosahedron, Paul, Trench, Trübner Co.Kim, H. (2000) ‘Functoriality for the exterior square of GL4 and symmetric

fourth power of GL2’, preprint.

Kim, H. and Shahidi, F. (2001a) ‘Functorial products for GL2 x GL3’, Ann.ofMath., to appear.

Kim, H. and Shahidi, F. (2001b) ‘Cuspidality of symmetric powers with applications’, Dztke Math. Jour, to appear.

Langlands, R. (1980) Base Changefor GL(2), Ann. Math. Studies, 96, Princeton University Press.

Labesse, 1-P. and Langlands, R. (1979) ‘L—indistinguishability for $L(2)’,Can. .1. Math. 31, 726—785.

Luo, W. (2001) ‘Non—vanishing of L—values and the strong Weyl law’,Ann. ofMath., to appear.

Phillips, R. and Sarnak, P. (1985) ‘On cusp forms for cofinite subgroups ofPSL(2, R)’ Invent. Math. 80, 339—364.

Sarnak, P. (2001). Letter to H. Kim and F. Shahidi, May 2001.Serre, 1-P. (1977) ‘Modular forms of weight one and Galois representations’.

In Algebraic Number Fields, A. Frölich (ed.), Academic Press, 193—268.

i-

126 Peter Sarnak

Acknowledgement I would like to thank S.D. Miller and F Shahidi for theirinsightful comments on my letter Samak (2001).

Maass Cusp forms with hiteger Coefficients

ShaIüdi, F. (1990) ‘AutomorphiC L—functions — a survey’. In Automorphic

forms, Shimura Varieties and L-functions, I, L. Clozel and J.S. Mime teds.),

Academic Press, 415—437

M. (2000) Diophantine Approximation on Linear Algebraic

Groups, SpringerVerlag.

Well, A. (1980) ‘On a certain type of characters of the idële-class group of an

algebraic number-field’. Collected Works II, Springer-Verlag, 255—261.

127

Ann. of