Geometric and p-adic Modular Forms of Half-Integral...

Geometric and p-adic Modular Forms of Half-Integral Weight

A thesis presented

by

Nicholas Adam Ramsey

to

The Department of Mathematics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Mathematics

Harvard University

Cambridge, Massachusetts

May 2004

c©2004 - Nicholas Adam Ramsey

All rights reserved.

Thesis advisor Author

Barry Mazur Nicholas Adam Ramsey

Geometric and p-adic Modular Forms of Half-Integral Weight

AbstractWe set up a geometric theory of modular forms of half-integral weight. This theory is equipped with

all of the standard features of the corresponding theory for integral weight, such as q-expansions, a

q-expansion principle, some base-change compatibilities, and geometrically defined Hecke operators.

To do this, we identify the space of modular forms of weight k/2 (k odd) and level 4N with

a certain space of rational functions on the modular curve X1(4N). Namely, we define a divisor

Σ4N,k on the curve X1(4N)/R for any Z[1/(2N)]-algebra R, and identify the space of modular forms

over R of weight k/2 and level 4N with the space of rational function on X1(4N)/R whose poles are

no worse than that specified by the divisor Σ4N,k. The caveat is that Σ4N,k generally has nonintegral

(rational) coefficients.

To construct the Hecke operators, we use the Hecke correspondences on the curve X1(4N)

obtained from the usual pair of degeneracy maps X1(4N ; m) X1(4N). To construct a Hecke

operator out of such a correspondence requires a rational function on X1(4N ; m) satisfying a certain

order of vanishing requirement. In the “good” case in which gcd(4N, m) = 1, we prove that no

such function exists unless m is a perfect square, in which case such a function exists and is unique

up to a constant multiple. This is the geometric explanation of the well-known fact for classical

modular forms of half-integral weight that there are no (nonzero) good Hecke operators of non-

square index. We prove that for m = p2, p 6 |2N , our Hecke operator Tp2 agrees with the classical

one when (R = C) by showing that it has the expected effect on q-expansions at the cusp ∞. We

also construct a number of Hecke operators in cases where m is not prime to 4N , and show that

these also agree with the classical analogs.

We also consider Kohnen’s +-space in our setting. To define this space in the classical

setting one simply requires one half (depending on the weight) of the Fourier coefficients to vanish.

We offer evidence that this definition is still sensible in our more general setting by proving that

an identification between this space and an eigenspace of a certain natural operator persists in this

generality.

After the geometric theory is set up, we use it to define p-adic modular forms of half-integral

weight. We construct the good Hecke operators on the spaces of such forms. In addition, in the

“sufficiently overconvergent” case, we construct the operator Up2 and prove that it is a completely

continuous endomorphism of the space of overconvergent modular forms. This implies that Up2 has

a Fredholm theory and opens the door for analogs of a number of the interesting results of Coleman,

Katz, Mazur, and others, on modular forms of integral weight. For example, one should be able to

iii

Abstract iv

construct an eigencurve parameterizing all overconvergent eigenforms forms of finite slope.

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Preliminaries 1

1.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Tate curve and q-expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Modular Forms of Half-Integral Weight 8

2.1 Holomorphic Modular Forms of Half-Integral Weight . . . . . . . . . . . . . . . . . . 82.2 Algebraization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Modular Forms of Half-Integral Weight . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 q-Expansions and the q-Expansion Principle . . . . . . . . . . . . . . . . . . . . . . . 122.5 Changes-of-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 The Modular Unit Θm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7.1 The operators Tm, gcd(4N, m) = 1 . . . . . . . . . . . . . . . . . . . . . . . . 222.7.2 The operators Up2 , p odd, p|N . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.3 The operators Up, p odd, p|N . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.4 The operator U4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8 The operator W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.9 Kohnen’s +-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 p-adic Modular Forms of Half-Integral Weight 41

3.1 Overconvergent p-adic Modular Forms of Half-Integral Weight . . . . . . . . . . . . . 413.2 The situation when p is nilpotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Good p-adic Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 The Canonical Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 The operator Up2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Bibliography 55

v

Acknowledgments

I would like to thank my advisor Barry Mazur for suggesting a problem which led me

to this work, for numerous conversations and insightful remarks, and for his unwavering optimism

throughout. I have also greatly benefited mathematically from conversations and interaction with

Izzet Coskun, Noam Elkies, Grigor Grigorov, and Joe Harris.

In addition to those already mentioned, I would like to thank Ben Howard, John Mackey,

Jerrel Mast, Ben Peirce, and Rob Neel for their integral contributions to my sometimes all-too-

extensive non-mathematical life while at Harvard.

My experience at the Harvard Mathematics Department was enriched on a daily basis by

Irene Minder and Joe Harris. Their down-to-earth nature and interest in the lives of the students

in the department is unparalleled. In addition, they have been instrumental in some of the greatest

beverage innovations in the history of the Harvard Mathematics Department.

I would like to thank my family for their encouragement of my educational goals, and for

putting up with me in some of my less favorable periods before I had any. I would also like to thank

Alexia, for always being there for me.

There are a number of people who went out of their way to help me at earlier stages of

my education, and it is my great pleasure to extend a special thanks to them now. I thank Robert

Kottwitz, Arunas Liulevicius, Diane Herrmann, Matam Murthy, and Paul Sally for providing a

warm and rich environment in which to develop as an undergraduate. I thank the Mathematics

Department of Bradley University, particularly Gerald Jungck and Michael McAsey, for giving me

a chance and subsequently devoting more of their time than ever could have been asked of them to

me. Finally, I thank Paula Davis, Karen Smith, and especially Tom Wade. I cannot overstate the

effect that the tremendous efforts of these three people had on my life.

vi

Chapter 1

Preliminaries

1.1 Introduction and Motivation

The aim of this article is to develop a geometric theory of modular forms of half-integral

weight; that is, to realize spaces of half-integral weight modular forms as sections of certain bundles

over moduli spaces. Looking at the corresponding theory for integral weight forms, one sees that

such a theory, if it is to be of arithmetic interest, should include the following.

• The moduli spaces and bundles involved should have integral models (say over Z[1/N ] for some

integer N depending on, e.g. level structure imposed),

• a notion of q-expansions which agrees with the existing one for holomorphic modular forms,

• a q-expansion principle which states that one can read off from its q-expansion over what ring

one’s modular form is defined

• some basic change-of-ring compatibilities, and

• geometrically defined Hecke operators which agree with the classical ones over C.

We achieve this by identifying the space of modular forms of half-integral weight with

a certain space of rational functions on the moduli space of elliptic curves with the appropriate

level structure. After this geometric theory is set up, we exploit it to introduce a p-adic theory of

modular forms of half-integral weight which includes the above forms as a “classical” subspace. The

theory is equipped with the notion of overconvergence and a p-adic Up2 operator, which is completely

continuous in the overconvergent case. This opens the door for a spectral theory as in [7] and [3].

1

Chapter 1: Preliminaries 2

1.2 Conventions

We choose once and for all a square root i ∈ C of −1. With this fixed, it makes sense to

impose the convention that all (nonzero) square roots are taken to have argument in the interval

[−π/2, π/2), and we do so for the duration of the paper.

Our choice of i furnishes an upper half plane h consisting of complex numbers with positive

imaginary parts. For τ ∈ h we define the theta functions with characteristics as the collection of

functions

ϑ(τ, α, β) =∑

n∈Z

eπi(n+α)2τ+2πi(n+α)β

for rational numbers α and β (the characteristics). These functions obey a well-known transformation

formula under the group SL2(Z), namely, for any

(a b

c d

)∈ SL2(Z),

we have

ϑ

(aτ + b

cτ + d, α′, β′

)= µ(cτ + d)1/2ϑ(τ, α, β),

where (α′

β′

)=

(d −c

−b a

)(α

β

)+

1

2

(cd

ab

)

and µ is an eighth root of unity depending only on a, b, c, d, α, β. This formula can be found in [4].

The root of unity µ is hard to control in general, but for the standard generators

(1 1

0 1

)and

(0 −1

1 0

)

of SL2(Z) one can pin µ down precisely. For the details, which we will use freely in the sequel, see

[12].

We define another theta function by θ(τ) = ϑ(2τ, 0, 0) =∑

n∈Ze2πin2τ . This function

obeys a transformation formula under the congruence subgroup Γ0(4), namely

θ

(aτ + b

cτ + d

)= ε−1

d

( c

d

)(cτ + d)1/2θ(τ) for

(a b

c d

)∈ Γ0(4),

where for odd d we define

εd =

1 d ≡ 1 (mod 4)

i d ≡ 3 (mod 4).

For more details on this function, including conventions for the Legendre symbol appearing in the

above formula, see [13] or [9].


For a positive integer N , we denote by Y1(N)an the Riemann surface Γ1(N)\h and by

X1(N)an its compactification (as given in [14], for example). In addition, Y1(N) and X1(N) will

denote the moduli schemes for the usual Γ1(N) and generalized Γ1(N) problems, respectively, when

these spaces exist. When these problems are not representable, we reserve these names for the

corresponding coarse moduli schemes. All of this can be found in [8]. We will always choose to

uniformize these problems over C via the map

Γ1(N)\h −→ Y1(N)(C)

τ 7−→

(Eτ ,

1

N

)

where Eτ denotes the elliptic curve C/〈1, τ〉.

For a pair of integer N and M , we will also consider the moduli problem classifying triples

(E, P, C) consisting of

• an elliptic curve E,

• a point P of exact order N , and

• a cyclic subgroup of order M not meeting the subgroup generated by the point P .

The corresponding moduli space will be called Y (N ; M), and X(N ; M) will denote its compactifica-

tion classifying generalized elliptic curves with the appropriate versions of the above level structures.

For any N and M we can uniformize this problem over C via the map

(Γ1(N) ∩ Γ0(M))\h −→ Y (N ; M)(C)

τ 7−→

(Eτ ,

1

N,⟨ τ

M

⟩)

where

Γ0(M) =

(a b

c d

)∈ SL2(Z)

∣∣∣∣∣ M |b

.

When gcd(N, M) = 1 we can also uniformize via the map

(Γ1(N) ∩ Γ0(M))\h −→ Y (N ; M)(C)

τ 7−→

(Eτ ,

1

N,

⟨1

M

⟩).

Both uniformizations will be used in what follows.

We now detail our conventions on the Tate curve, which differ from the standard ones (as

found in [7]), for example. See Section 1.3 for more information on the Tate curve. Let Tate(q)

denote the Tate curve over the base Z((q)) = Z[[q]][1/q] as in Section 1.3 (see also [7] and [8]). When


in the presence of level N structure, we introduce a formal variable qN (thought of as q1/N ) and an

inclusion of rings

Z((q)) → Z((qN ))

q 7−→ qNN .

In practice, we simply identify q and qNN . More generally, for M |N , we define qM = q

N/MN . Note

that the N -torsion on Tate(q) is now defined over Z((qN ))⊗ Z[ζN ], and in fact is

ζiNqj

N, 0 ≤ i, j ≤ N − 1.

The result of this normalization is that the q-expansions of algebraic modular forms gotten

by evaluating them at pairs (Tate(q), P ), with P a point of order N , look like the classical ones. By

this it is meant that they occur in the variable qN , whereas the classical ones occur in e2πiτ/N . In

the sequel, we will therefore also refer to this function of τ as qN . Of course, both the classical and

geometric q-expansion will generally be expansions in qM for some divisor M of N , depending on

the width of the cusp associated to (Tate(q), P ).

There are generally a number of points P of order N on Tate(q) giving rise to a particular

cusp on X1(N), and the corresponding q-expansions gotten by evaluating at (Tate(q), P ) for these

various P will be different. Let R be a Z[1/N ]-algebra and let F be a rational function on X1(N)/R.

We define the q-order of F at (Tate(q), P ) to be the rational number a/N such that the leading

term of

F (Tate(q), P ) ∈ Z((qN ))⊗R[ζN ]

is of degree a in qN . If N > 4, so that X1(N) is a fine moduli scheme, then the q-order of F at

(Tate(q), P ) is the actual order of vanishing of F at the cusp associated to (Tate(q), P ) divided by

the width of this cusp. In particular, we see that the q-order depends only on this cusp.

The notion of q-order and the above relationship to actual order of vanishing extends

easily to the spaces X(N ; M), as long as they are fine moduli schemes. The only space which we

will have occasion to consider in what follows which is not fine is the scheme X1(4). The generalized

Γ1(4) problem is not representable, as one of the cusps (that is, the object in the moduli problem

corresponding to one of the cusps) has an automorphism of order two (see [6]). The image of this

cusp on the coarse moduli scheme X1(4) will be denoted by 1/2 in what follows, and has width 1.

The q-order of a rational function on X1(4) at a level structure corresponding to the cusp 1/2 is the

actual order of vanishing there divided by 2.


1.3 The Tate curve and q-expansions

If we fix a choice of uniformization

Γ\h∼−→ Y1(N)(C),

for some subgroup Γ of SL2(Z), then a meromorphic function F (τ) on X1(N)an which is holomorphic

outside the cusps gives rise to a rational function on the curve X1(N)/C. This, in turn, can be

interpreted as a rule (also called F ) which takes an elliptic curve over a C-algebra S and point of

order N and associates to it an element of S subject to the usual compatibilities under isomorphisms

and base changes. In this section, we explain how one computes F (Tate(q)/C, P ) for the various

points P of order N on Tate(q)/C.

We begin with some more background on the Tate curve. We define Tate(q) to be the

elliptic curve over Z((q)) defined by the cubic equation

y2 + xy = x3 + B(q)x + C(q)

where

B(q) = −5

∞∑

n=1

σ3(n)qn

C(q) =

∞∑

n=1

(−5σ3(n)− 7σ5(n)

12

)qn

and σk(n) denotes the sum of the kth powers of the (positive) divisors of n. If we put q = e2πiτ in

this definition we get an elliptic curve over C which we will denote Tateτ and the map

C×/〈q〉 −→ Tateτ

t 7−→ (x(t), y(t))

where

x(t) =∑

k∈Z

qnkt

(1− qnkt)2− 2

∞∑

k=1

qnk

1− qnk

y(t) =∑

k∈Z

(qnkt)2

(1− qnkt)3+

∞∑

k=1

qnk

1− qnk

is an isomorphism of elliptic curves (see Appendix A1.2 of [7]). Moreover, note that one has the

isomorphism

Eτ = C/〈1, τ〉∼−→ C×/〈q〉

z 7−→ e2πiz


The result is that, intuitively, the curve Eτ is a base-change of Tate(q) under the spe-

cialization q 7→ e2πiτ . Literally this makes no sense as this does not extend to a homomorphism

Z((q))→ C. To remedy this, we let M denote the subring of Z((q)) consisting of the q-expansions of

meromorphic functions in the disk |q| < 1 which are holomorphic on the punctured disk 0 < |q| < 1

(and have integral q-expansions). Since B(q) and C(q) belong to M, the curve Tate(q) is actually

defined over M. More generally, for each integer n ≥ 1, we define Mn to be the same thing as M

with q replaced by the variable qn. For n|m, we identify Mn with a subring of Mm via qn 7→ qm/nm .

Let ζN denote the point on Tate(q) given by setting t = e2πi/N in the formulas for x(t)

and y(t) above (where q is now simply the variable q, not e2πiτ ). Also, let qN denote the point on

Tate(q) given by setting t = qN ∈ Mn in these formulas. Then the N -torsion on Tate(q) consists

exactly of the N2 points

ζiN qj

N , 0 ≤ i, j ≤ N − 1

and in particular is all defined over Mn ⊗ Z[1/N, ζN ]. See Appendix A1.2 of [7] for more discussion

on this topic.

For each τ ∈ h, we have a family of homomorphisms

φτ : Mn −→ C

f(q) 7−→ f(e2πiτ/n)

which are compatible with the inclusions defined between the various Mn’s (which is why we suppress

the n form the notation φτ ). Thus upshot is that we now have isomorphism

Tate(q)×φτ Spec(C)∼−→ Eτ

as given above. Moreover, on the N -torsion, the above map is

ζiNqj

N 7−→i + jτ

N.

Let F be as in the beginning of this section, and let Tate(q)/C denote the extension of

scalars of Tate(q) to Z((q)) ⊗ C. We wish to determine the values F (Tate(q)/C, P ) for the various

points P of order N on Tate(q)/C. Since this data is defined over Mn ⊗ C (and we regard it as

such), all of these q-expansions are contained in this ring. Thus we can specialize qn via φτ and use

base-change compatibilities to write

φτ (F (Tate/C, ζiNqj

N )) = F (Tate(q)/C ×φτ Spec(C), P ×φτ Spec(C))

= F

(Eτ ,

i + jτ

n

).


To determine this value, we must use our chosen uniformization. For example, if the

uniformization is the usual one

SL2(Z)\h −→ Y1(N)(C)

τ −→

(Eτ ,

1

N

)

then to compute F (Eτ , (i + jτ)/N), we would find an element

(a b

c d

)∈ SL2(Z)

such that we have an isomorphism

(Eτ ,

i + jτ

N

)∼−→

(E aτ+b

cτ+d,

1

N

)

z 7−→z

cτ + d

so that, by compatibility with isomorphisms, we have

F

(Eτ ,

i + jτ

N

)= F

(aτ + b

cτ + d

).

In the text, we often use this procedure in the following suggestive shorthand:

F (Tate(q), ζiN qj

N )(τ) = F

(Eτ ,

i + jτ

N

)= F

(aτ + b

cτ + d

).

Moreover, the obvious modification to Γ(N ; M)-type level structures works in the same manner.

Chapter 2

Modular Forms of Half-Integral

Weight

2.1 Holomorphic Modular Forms of Half-Integral Weight

To define modular forms of half-integral weight we identify the classical holomorphic ones

with a certain space of rational functions on a modular curve using the classical theta function

θ(τ) =∑

n∈Z

e2πin2τ

for τ in the upper half-plane h. Then we see that this identification makes good sense algebraically

over more general rings. For properties of classical holomorphic modular forms of half-integral

weight, see [13] or [9].

We follow the conventions in [13] in that holomorphic modular forms of weight k/2 (k odd)

are those that transform like θk. Thus, for a positive integer N , we define a holomorphic modular

form of weight k/2 and level 4N to be a holomorphic function f on h satisfying

f

(aτ + b

cτ + d

)= ε−k

d

( c

d

)k

(cτ + d)k/2f(τ), for all

(a b

c d

)∈ Γ1(4N)

and some additional growth conditions at the cusps. The space of such forms is denoted by

Mank/2(4N). The growth conditions above are a little more subtle than those for modular forms

of integral weight, due to the plenitude of irregular cusps. See [13] or [9] for the details.

An element f ∈ Mank/2(4N) is of the form f = F · θk for some meromorphic function F

on h which is invariant under the action of Γ1(4N). That is, F is a meromorphic function on the

Riemann surface Y1(4N)an = Γ1(4N)\h. We wish to determine for which such F is the resulting

8

Chapter 2: Modular Forms of Half-Integral Weight 9

form f holomorphic. It follows from [12], Chapter 1, that θ is nonvanishing in h, so that we first

require F to be holomorphic on Y1(4N). It remains to see what conditions must be placed on F at

the cusps.

To do this, we must examine the behavior of θ at the cusps of X1(4)an. There are three

such cusps and one finds via the formulas for transforming “theta functions with characteristics” in

section 1.2 that the q-expansions of θ are given as follows (defined up to a constant except in the

first case)

cusp on X1(4)an q − expansion

∞ θ1(q) :=∑

n∈Zqn2

1/2 θ3(q) := q4

∑n∈Z

qn2+n

0 θ2(q) :=∑

n∈Zqn2

4

(2.1)

where, once again, we use the notation qh = e2πiτ/h.

So θ is nonvanishing on cusps of X1(4N)an mapping to ∞ or 0 and we correspondingly

require that F be holomorphic at such cusps. Let c be a cusp mapping to the cusp 1/2 and let wc

be the width c. Note that wc coincides with the ramification index of the map X1(4N)an → X1(4)an

at c, as the cusp 1/2 on X1(4) has width 1. Thus we see that for Fθk to be holomorphic we must

have

ordcF +kwc

4≥ 0.

Summing up, we introduce the Q-divisor

Σ4N,k =∑

c∼1/2

kwc

4· c,

where the sum is over all cusps on X1(4N)an mapping to 1/2 on X1(4)an.

In general, for a divisor D with rational coefficients on a smooth curve X , the sheaf OX(D)

makes no sense. However, as a notational device, we define

H0(X, D) = s ∈ C(X) | (s) ≥ −D,

and similarly for a Q-divisor on any nice scheme (e.g. a smooth curve over an arbitrary base). If we

let ⌊D⌋ denote the divisor obtained from D by taking the floor of all of the coefficients, then since

the coefficients of (s) are integral, we have

H0(X, D) = H0(X, O(⌊D⌋)).

The point of keeping the divisor D as opposed to ⌊D⌋, or equivalently the sheaf O(⌊D⌋), is that

the latter do not behave well with respect to pulling back through maps. For example, if N |M and

π : X1(4M)an → X1(4N)an is the usual degeneracy map, then π∗Σ4N,k = Σ4M,k, whereas this fails

for the floor of Σ4N,k.


If χ : (Z/4NZ)× → C× is a character, then we may also identify the space Mank/2(4N, χ)

(defined in the usual way) with the space of F satisfying the additional condition

F

(aτ + b

cτ + d

)= χ(d)F (τ) for

(a b

c d

)∈ Γ0(4N).

Note that if we allow k to be even, then we have

Mank/2(4N, χ) =

Man

k/2(4N, χ ·(−1·

)) 2||k

Mank/2(4N, χ) 4|k

(2.2)

where the spaces on the right are given their usual integral-weight definitions.

2.2 Algebraization

The identification made above between Mank/2(4N) and H0(X1(4N)an, Σ4N,k) is a simple

consequence of knowing the “divisor” of θ on X1(4)an. Divisor is in quotes because this makes no

sense as θ has not been realized as the section of any bundle on X1(4)an. Indeed, this “divisor” as

it stands appears to have nonintegral coefficients.

We wish to do something similar on the algebraic curve X1(4)/C. This curve is the coarse

moduli scheme for the generalized Γ1(4) problem, and X1(4)an is its analytification. This problem

is not representable by a scheme, as one of the cusps has an automorphism of order 2 (in the sense

that the object in the moduli problem corresponding to this cups has an automorphism of order 2

– see [6]). It is representable by a smooth proper Deligne-Mumford stack and is equipped with a

natural line bundle ω which is the “push forward of the relative cotangent bundle from the universal

elliptic curve.” The bundle ω does not descend to X1(4), but its square ω2 does.

So, to remedy the situation we consider instead the function θ4. This is now an honest

modular form of weight 2 on X1(4)an and accordingly furnishes a section of ω2. By GAGA, it

corresponds to an algebraic section of ω2 on X1(4)/C and by the q-expansion principle (see [7]) is in

fact defined over Z[1/2]. Let 1/2 denote the cusp of X1(4) corresponding to (i.e. which is the image

of in the map from the stack to X1(4)) the pair

(Tate(q), ζ4q2),

which is defined over Z[1/2, ζ4][[q2]].

Proposition 1 The section θ4 above furnishes an isomorphism

ω2X1(4)/Z[1/2]

∼= OX1(4)/Z[1/2]([1/2]).


Proof. We first compute the q-expansions of θ4 in terms of the six level 4 structures on Tate(q). The

result is summarized in the following table (compare with 2.1).

point on Tate(q) cusp on X1(4) q − expansion

ζ4 ∞(∑

n∈Zqn2)4

ζ4q2 1/2 q(∑

n∈Zqn2+n

)4

q4 0 − 14

(∑n∈Z

qn2

4

)4

ζ4q4 0 − 14

(∑n∈Z

(ζ4q4)n2)4

ζ24q4 = −q4 0 − 1

4

(∑n∈Z

(−q)n2)4

ζ34q4 = −ζ4q4 0 − 1

4

(∑n∈Z

(ζ34q4)

n2)4

(2.3)

Thus, at least over Z[1/2, ζ4], the form θ4 gives an invertible section of ω2(−[1/2]) (with inverse the

section gotten by the above reasoning applied to θ−4). As this section and its inverse are defined

over Z[1/2] by the q-expansion principle (applied at ∞, for example), we get the desired result.

2.3 Modular Forms of Half-Integral Weight

The result of the previous section makes it sensible to identify, for any Z[1/2N ]-ring R and

odd integer k, the “space of modular forms of weight k/2 and level 4N over R” with the R-module

of sections

H0(X1(4)/R, Σ4N,k),

where

Σ4N,k =∑

c∼1/2

kwc

4· c

and the sum is taken over all cusps on X1(4N)/R mapping to the point 1/2 ∈ X1(4)/R (as defined

in the previous section). Indeed, we adopt this as a definition.

Definition 2 Let N be a positive integer and k an odd positive integer. For a Z[1/2N ]-ring R we

define the space of modular forms of weight k/2 to be

Mk/2(4N, R) := H0(X1(4N)/R, Σ4N,k).

Note that, since Σ4N,k is supported on the cusps, an element of H0(X1(4N)/R, Σ4N,k) can

be identified with a rule which associates to every pair (E/S , P ) consisting of an elliptic curve E over

an R-algebra S and a point P of order 4N , an element of S, subject to the usual compatibilities

and conditions on Tate curves specified by Σ4N,k. Explicitly, we require this rule to be insensitive

to isomorphisms (over S) of the data involved and compatible with arbitrary pullbacks of the data


under changes of R-ring. To a point P of order 4N on Tate(q), thought of as an elliptic curve over

R((q)), we require that the rule assign an element of q−e4NR[[q4N ]] where

e =4N

word(Tate(q),P )Σ4N,k

and w is the width of the cusp associated to (Tate(q), P ).

Definition 3 For a character χ : (Z/4NZ)× → R×, we define the subspace Mk/2(4N, χ, R) of

Mk/2(4N, R) to be that for which the associated rule F has the additional property that F (E, dP ) =

χ(d)F (E, P ) for all d ∈ (Z/4NZ)×.

Remark 4 If we had allowed even k in the above definitions, they would not agree with the usual

definition of integral weight modular forms, the difference being exactly as in (2.2).

2.4 q-Expansions and the q-Expansion Principle

Elements of H0(X1(4N)/R, Σ4N,k) have natural q-expansions gotten by evaluating the cor-

responding rule on Tate curves with level structure. These do not agree with the usual notion of

q-expansion of a holomorphic modular form of half-integral weight in the case R = C (and trans-

lating between the algebraic and analytic worlds via GAGA). To recover the classical q-expansions

one must multiply these naive q-expansions by that of the appropriate power of the corresponding

q-expansion of θ.

All of the level 4 structures on Tate(q) are defined over the ring Z((q4))⊗Z[1/2, ζ4], where

we have fixed some primitive fourth root ζ4. Examining the q-expansions in (2.3) we see that, up to

a root of unity, the q-expansions of θ are as follows.

level structure on Tate(q) q-expansion

ζ4

∑n∈Z

qn2

ζ4q2 q4

∑n∈Z

qn2+n

q41

1+ζ4

∑n∈Z

qn2

4

ζ4q41

1+ζ4

∑n∈Z

ζn2

4 qn2

4

ζ24q4 = −q4

11+ζ4

∑n∈Z

(−1)nqn2

4

ζ34q4 = −ζ4q4

11+ζ4

∑n∈Z

(−1)nζn2

4 qn2

4

Note that the last four level structures all correspond (that is, degenerate) to the same cusp on

X1(4). Also, the root of unity ambiguity is not an issue, as these q-expansion (excepting the one at

∞) are only well-defined up to a root of unity even in the classical setting (see [9]).

Definition 5 Let F ∈Mk/2(4N, R) and let P be a point of order 4N on Tate(q). In the table above,

let θP denote the q-expansion of θ at the image (Tate(q), P ) on X1(4) under the degeneracy map

(E, Q) 7→ (E, NQ). The q-expansion of F at (Tate(q), P ) is defined to be

F (Tate(q), P )θkP ∈ Z((q4N ))⊗Z R[ζ4N ].


Remark 6 The condition on F at the cusps exactly ensures that the q-expansions of F are in fact

all contained in R[[q4N ]]. Indeed, for P mapping to ζ4q2, suppose that F (Tate(q), P ) = qd4N (a0 +

a1q4N + · · · ) with a0 6= 0. Then we have required that

d ≥ −4N

w·kw

4= −Nk,

so that the power of q4N occurring the q-expansion of F is d+Nk ≥ 0. The result is obvious for the

other level structures.

We now come to the q-expansion principle. If we were using the naive q-expansions gotten

by simply evaluating our rules at Tate curves with level structures, then the q-expansion principle

would follow the integral weight version verbatim (see [7]) as it is a purely geometric statement and

depends only on the nonsingularity and irreducibility of the curves X1(4N). It is easily adapted to

our q-expansions.

As in the integral-weight case, we must first define modular forms over modules (as opposed

to rings). Let K be any Z[1/(2N)]-module. We define a modular form of weight k/2 and level 4N

to be a rule which assigns to any pair (E/R, P ) as before where R is a Z[1/(2N)]-algebra, an element

of R⊗Z[1/(2N)] K, subject to the usual compatibilities and conditions on Tate curves. Note that, in

case K is actually a Z[1/(2N)]-algebra, this spaces agrees with the one previously defined (that is,

there is a simple canonical identification between the two). We will therefore denote this space of

forms by Mk/2(4N, K) as well.

Note that, in contrast to the case of a ring, an element of Mk/2(4N, K) has q-expansions

in

Z((q4N ))⊗Z Z[1/(2N), ζ4N ]⊗Z[1/(2N)] K.

In particular, when K is a ring, this will not in general agree with the notion of q-expansion defined

above. Nonetheless, it is precisely for these q-expansions that we have a q-expansion principle.

Theorem 7 Let L ⊂ K be an inclusion of Z[1/(2N)]-modules and let F ∈Mk/2(4N, K). Let P be

point of order 4N on Tate(q) and suppose that the q-expansion of F at (Tate(q), P ) has coefficients

in Z[1/(2N), ζ4N ]⊗Z[1/(2N)] L. Then F ∈Mk/2(4N, L).

Proof. Referring to the table above, we see that each θP and θ−1P have coefficients in Z[1/2, ζ4]. It

follows that the q-expansion of F at (Tate(q), P ) has coefficients in Z[1/(2N), ζ4N ] ⊗Z[1/(2N)] L if

and only if F (Tate(q), P ) does. As observed above, this is sufficient to see that F itself comes from

Mk/2(4N, L), which is done by applying the q-expansion principle in weight 0 (the poles are of no

consequence).


2.5 Changes-of-ring

Any map R→ S of Z[1/(2N)]-algebras induces a map

Mk/2(4N, R)⊗R S −→Mk/2(4N, S).

We would like conditions under which this map is an isomorphism of S-modules. As these are

sections of the sheaf O(⌊Σ4N,k⌋) on the respective spaces, it suffices by standard base-changing

theorems to prove that

H1(X1(4N)/R, O(⌊Σ4N,k⌋)) = 0

(just as in [7]).

Theorem 8 This vanishing of H1 holds (for any R) if k ≥ 4.

Proof. By Serre duality it suffices to show that

deg(⌊ΣN,k⌋) > deg(Ω1X1(4N)/R).

Pulling back the isomorphism of Proposition 1 through the degeneracy map

X1(4N) −→ X1(4)

(E, P ) 7−→ (E, NP )

we see that

ω2X1(4N)

∼= O(Σ4N,4).

By the Kodaira-Spencer isomorphism, we have

Ω1X1(4N)

∼= ω2X1(4N)(−C)

where C is the divisor of cusps. Thus

deg(Ω1X1(4N)) = deg(ω2

X1(4N))− |C| < deg(Σ4N,4) ≤ deg(⌊Σ4N,k⌋)

whenever k ≥ 4.

Remark 9 In weights 1/2 and 3/2 this inequality does not hold in general. With some help from

MAGMA (see [1]), one finds that the first counterexample in weight 1/2 occurs at level 4 · 5, and the

first counterexample in weight 3/2 occurs at level 4 · 17. The author does not know if base change

holds in these cases.


2.6 The Modular Unit Θm

In order to construct the Hecke operators geometrically in half-integral weight, we must

introduce a number of special rational functions on modular curves of the form X(4N, m).

For a fixed odd integer m > 1, consider the holomorphic function on h

Θm(τ) :=θ(mτ)

θ(τ).

If

γ =

(a b

c d

)∈ Γ1(4) ∩ Γ0(m)

we have

θ

(m

aτ + b

cτ + d

)= θ

(a(mτ) + bm

(c/m)(mτ) + d

)

=

(c/m

d

)ε−1

d ((c/m)(mτ) + d)12 θ(mτ)

=( c

d

)(m

d

)ε−1

d (cτ + d)12 θ(mτ)

and

θ

(aτ + b

cτ + d

)=( c

d

)ε−1

d (cτ + d)12 θ(τ).

Thus Θm gives a regular function on the Riemann surface

Y (Γ1(4) ∩ Γ1(m))an = Y1(4Nm)an

with Nebentypus character (m· ). In particular, note that Θm descends to Y (Γ1(4)∩ Γ0(m)) exactly

when m is a square.

We must examine the behavior at the cusps. Recall that θ is nonvanishing at cusps whose

image in X1(4)an is distinct from 1/2. Moreover, it is easy to see that an r/s ∈ Q in lowest terms

is equivalent to 1/2 under Γ1(4) if and only if 2||s. In particular this is true of r/s if and only if it

is true of m(r/s). So the divisor of Θm is supported on the cusps of X1(4m)an mapping to 1/2.

These cusps are represented by the level structures ζi4mqj

4m on Tate(q) with i odd and 2||j.

If

γ =

(a b

c d

)∈ SL2(Z)

takes ∞ to this cusp, then (c, d) ≡ (j, i) (mod 4m). We wish to compute Θm|γ, or at least its order

of vanishing. Noting that 2||c, we let 2h = gcd(j, 4m) = gcd(c, 4m) (so this cusp has width 2m/h,

assuming m > 1).

Thus gcd(m/h, c/2h) = 1 and we can find an integer x with

d−c

2hx =

m

hl


for some l ∈ Z, that is, with

2dh− cx

divisible by 2m. Now we let m′ = m/h and rewrite

2maτ + b

cτ + d=

am′(

2hτ+xm′

)− ax + 2bh

c2h

(2hτ+x

m′

)+ 2dh−cx

2m

and check that the matrix (am′ −ax + 2bh

c/(2h) (2dh− cx)/(2m)

)

is in SL2(Z).

So we are faced with computing

ϑ

(am′

(2hτ+x

m′

)− ax + 2bh

c2h

(2hτ+x

m′

)+ 2dh−cx

2m

)

and

ϑ

(a(2τ) + 2b

(c/2)(2τ) + d

)

It suffices for our purpose to evaluate these only up to a constant multiple. We use the

general formula in Section 1.2 for transforming theta functions with characteristics. Moreover, in

all our computations, α′ = β′ = 0, so

(α

β

)=

1

2

(a c

b d

)(cd

ab

)∈

1

2Z2.

Note that, in this case, ϑ only depends on α and β mod 1 (up to a root of unity).

One verifies that α = 1/2 and β = 0 mod 1 for the two matrices with which we are

concerned, and that

ϑ(τ, 1/2, 0) = e2πiτ/8∑

n∈Z

eπi(n2+n)τ .

Thus we see that the unit Θm has q-order

2h

8m′−

2

8=

h2 −m

4m

and therefore actual order of vanishing

2m

h

h2 −m

4m=

h2 −m

2h

(which is an integer).

Since the q-expansion of Θm at ∞ is

∑qmn2

∑qn2 ∈ Z[[q]],


we see by GAGA and the q-expansion principle that Θm is in fact an algebraic function defined over

Z[1/2m].

Now let us specialize to the case m = p2, p and odd prime. We will need a number of

different q-expansions of Θp2 in terms of Tate curves when we consider the Hecke operators Tp2 .

One easily sees that the cyclic subgroups of order p2 of Tate(q) are

〈ζp2〉, 〈ζip2qp2〉, 〈ζj

p2qp〉, 0 ≤ i < p2, 1 ≤ j < p.

Note that we can, of course, modify i and j by multiples of p2 and p, respectively, instead of using

the residues shown. This will be useful in the sequel when we will find it convenient to assume, e.g.,

that j is 1 mod 4.

Forming the quotient by these groups, one gets

Tate(q)/〈ζp2 〉∼−→ Tate(qp2

)

t 7→ tp2

Tate(q)/〈ζip2qp2〉

∼−→ Tate(ζi

p2qp2)

t 7→ t

Tate(q)/〈ζjp2qp〉

∼−→ Tate(ζj

pq)

t 7→ tp

Only the last of the above three does not work quite the same as in the case of subgroups of order

p. The map is most easily seen as a composite

Tate(q)/〈ζjp2qp〉

t7→tp

−→ Tate(qp)/〈ζjpq〉

t7→t−→ Tate(ζj

pq).

To evaluate Θp2 at the various Tate curves we exploit the usual yoga whereby one pretends

that one can specialize Tate(q) to the curves Eτ = C/〈1, τ〉 and use base-change compatibilities to

reduce to the analytic functions used to define Θp2 . For the details on this technique, see Section

1.3.

Θp2(Tate(q), ζ4, 〈ζp2〉)(τ) = Θp2

(Eτ ,

1

4,

⟨1

p2

⟩)

=θ(p2τ)

θ(τ)=

∑qp2n2

∑qn2

Note that (Tate(q), ζ4, 〈ζip2qp2〉) arises from (Tate(q), ζ4, 〈qp2〉) via the change-of-rings qp2 7→

ζip2qp2 . We compute

Θp2(Tate(q), ζ4, 〈qp2〉)(τ) = Θp2

(Eτ ,

1

4,

⟨τ

p2

⟩)


Choose k with p2 = 1 + 4k and note that the matrix

(1 −1

−4k p2

)

gives an isomorphism

(Eτ ,

1

4,

⟨τ

p2

⟩)∼−→

(E τ−1

−4kτ+p2,1

4,

⟨1

p2

⟩)

z 7−→z

−4kτ + p2

Thus

Θp2

(Eτ ,

1

4,

⟨τ

p2

⟩)=

θ(p2 τ−1

−4kτ+p2

)

θ(

τ−1−4kτ+p2

) =θ(

p2(τ/p2)−1−4k(τ/p2)+1

)

θ(

τ−1−4kτ+p2

)

=(−4k(τ/p2) + 1)1/2θ(τ/p2)

(−4kτ + p2)1/2θ(τ)

=1

p

∑qn2

p2

∑qn2 .

In general we have

Θp2(Tate(q), ζ4, 〈ζip2qp2〉) =

1

p

∑ζin2

p2 qn2

p2∑qn2 .

Finally, we tackle

Θp2(Tate(q), ζ4, 〈ζjp2qp〉)(τ) = Θp2

(Eτ ,

1

4,

⟨j

p2+

τ

p

⟩)

Let

γ =

(a b

c d

)

be any element of SL2(Z) which, under the corresponding isomorphism, preserves 1/4 but takes

j/p2 + τ/p to 1/p2. The first condition says that γ ∈ Γ1(4). The second says there exist integers

n, m such thatjp2 + τ

p

cτ + d=

1

p2+ n + m

aτ + b

cτ + d.

This, in turn, implies that d ≡ j (mod p2) and c ≡ p (mod p2). We note that all these conditions

ensure that (pa bp− aj

cp

dp−cjp2

)∈ Γ1(4)


and we compute

Θp2

(Eτ ,

1

4,

⟨j

p2+

τ

p

⟩)

=θ(p2 aτ+b

cτ+d

)

θ(

aτ+bcτ+d

)

=θ(

pa(τ+j/p)+bp−aj(c/p)(τ+j/p)+(dp−cj)/p2

)

θ(

aτ+bcτ+d

)

=ε−1(dp−cj)/p2

(c/p

(dp−cj)/p2

)((c/p)(τ + j/p) + (dp− cj)/p2)1/2θ(τ + j/p)(

cd

)(cτ + d)1/2θ(τ)

The numerator giveth

ε−1(dp−cj)/p2

(c/p

(dp− cj)/p2

)((c/p)(τ + j/p) + (dp− cj)/p2)1/2θ(τ + j/p)

= ε−1p

( c

d

)(p

j

)(cτ + d

p

)1/2

θ(τ + j/p)

and the denominator taketh away ( c

d

)(cτ + d)1/2θ(τ)

and that which remains is

ε−1p

p1/2

(j

p

)θ(τ + j/p)

θ(τ)=

ε−1p

p1/2

(j

p

) ∑ζjn2

p qn2

∑qn2

where we now assume that the j’s are congruent to 1 (mod 4) to simplify life.

2.7 Hecke Operators

Let X be a curve over an arbitrary base ring and F an invertible sheaf on X . Let

Yπ1

~~~~~~

~~~

π2

@@@

@@@@

X X

be a correspondence with finite and flat maps πi and let Θ a section of

π∗2F ⊗ π∗

1F∨.

This data gives a map

H0(X, F) −→ H0(X, F)

f 7−→ C · π2∗(π∗1f ·Θ)

(2.4)


for each constant C. This is exactly how the Hecke operators are constructed for integral weight

forms, in which case X = X1(N), Y = X1(Γ1(N) ∩ Γ0(p)), F = ωk, and Θ = 1 (which makes sense

as one has a canonical isomorphism π∗1ωk ∼= π∗

2ωk in the relevant cases).

In the case of half-integral weight modular forms, we do not have a sheaf, but rather a

Q-divisor on X . Nonetheless, something similar still works. In general, if D is a Q-divisor on X and

Θ is a rational function on Y with

(Θ) ≥ π∗2(−D)− π∗

1(−D) = π∗1D − π∗

2D,

we may define an endomorphism of H0(X, D) exactly as in (2.4) above. This follows from the simple

observation that if h is a rational function on Y , then (h) ≥ π∗2(−D) implies π2∗(h) ≥ −D. To see

this, note that (h) ≥ π∗2(−D) implies that (h) ≥ −⌊π∗

2D⌋. But this divisor has integral coefficients,

so this implies

(π2∗h) ≥ −π2∗⌊π∗2D⌋ ≥ −D,

as desired.

To construct Hecke operators, we pick an auxiliary integer m and define the curve X(4N ; m)

to be the (compactified) moduli space parameterizing triples (E, P, C) consisting of an elliptic curve,

a point P of order 4N , and a cyclic subgroup C of order m not meeting the subgroup generated by

P . This curve comes equipped with two degeneracy maps given by

π2 : X(4N ; m) −→ X1(4N)

(E, P, C) 7−→ (E, P )

π1 : X(4N ; m) −→ X1(4N)

(E, P, C) 7−→ (E/C, P/C).

To construct the Hecke operators, we seek a rational function Θ on X(4N ; m) with divisor

at least

π∗1Σ4N,k − π∗

2Σ4N,k.

That is, the divisor π∗1Σ4N,k−π∗

2Σ4N,k exactly specifies the minimal order of vanishing requirements

for the endomorphism associated to a rational function on Y be forced to preserve the space of

modular forms. Note that, since π1 and π2 have the same degree, the degree of this divisor is zero.

It follows that if Θ exists, it is unique up to a constant and has exactly this divisor (in particular,

this divisor must have integral coefficients).

In what follows, we will need to know the ramification indices of the maps π1 and π2 at a

cusp in terms of level structures on Tate(q) giving rise to the cusp. Note that the cyclic subgroups of

order m on Tate(q) are precisely the groups 〈ζlmqδ

m〉 where δ|m and l runs through a set of residues


modulo m/δ which are prime to δ (in particular, if gcd(δ, m/δ) > 1, this condition limits the residues

that occur).

Lemma 10 Let c be a cusp on X(4N ; m) and let (Tate(q), P, 〈ζlmqδ

m〉) be a level structure giving

rise to c. Let w denote the width of c and wi the width of πi(c), i = 1, 2. The the ramification

indices are given by

ecπi =

δ2

mww1

i = 1

ww2

i = 2.

Proof. Since c has width w, the triple (Tate(q), P, 〈ζlmqδ

m〉) is defined over Z((qw))⊗Z[1/(2Nm), ζ4Nm]

and arises from a unique map

Spec(Z((qw))⊗ Z[1/(2Nm), ζ4Nm]) −→ X(4N ; m). (2.5)

Note that

π2(Tate(q), P, 〈ζlmqδ

m〉) = (Tate(q), P ).

This pair arises from a unique map

Spec(Z((qw2 ))⊗ Z[1/(2N), ζ4N ]) −→ X1(4N).

Thus we have a commutative diagram

Spec(Z((qw))⊗ Z[1/(2Nm), ζ4Nm]) //

X(4N ; m)

π2

Spec(Z((qw2 ))⊗ Z[1/(2N), ζ4N ]) // X1(4N)

where by uniqueness the vertical map on the right arises from the natural inclusion of rings via

qw2 7→ qw/w2w . As the horizontal maps arise from completion along the associated cusps, is follows

that the ramification index is w/w2.

Now we turn to the more complicated π1. The conditions on l and δ imply that µδ ⊂

〈ζlmqδ

m〉, so that we may first take the quotient by this smaller group to obtain

(Tate(q)/µδ, P/µδ, 〈ζlmqδ

m〉)∼−→ (Tate(qδ), δP, 〈ζl

m/δqδm/δ〉)

t 7−→ tδ

Now we have 〈qδ〉 ⊂ 〈ζlm/δq

δm/δ〉 so that we can further quotient and get

(Tate(qδ)/〈ζlm/δq

δm/δ〉, δP/〈ζl

m/δqδm/δ〉)

∼−→ (Tate(ζl

m/δqδm/δ), δP )

t 7−→ t


This pair is the base change of (Tate(q), P ′) (for some point P ′) through the map

Z((qw1 ))⊗ Z[1/(2N), ζ4N ] −→ Z((qw1m/δ))⊗ Z[1/(2Nm), ζ4Nm/δ]

qw1

φ7−→ ζl

w1m/δqδw1m/δ

Thus the corresponding diagram of points on moduli spaces is

Spec(Z((qw)) ⊗ Z[1/(2Nm), ζ4Nm]) //

rr

X(4N ; m)

Spec(Z((qw1m/δ)) ⊗ Z[1/(2Nm), ζ4Nm/δ ])φ

// Spec(Z((qw1)) ⊗ Z[1/(2N), ζ4N ]) // X1(4N)

where we would like to argue that we can fill in the dotted line (by uniqueness) with the natural

inclusion

qw1m/δ 7→ qw

w1

δm

w

between the two rings. This makes no sense, as the exponent here is not generally an integer. This

problem is an artifact of having chosen too large a ring Z((qw1m/δ))⊗Z[1/(2Nm), ζ4Nm/δ] (the pair

(Tate(ζlm/δq

δm/δ), δP ) is generally defined over a smaller ring), though even if we had chosen the

“right” ring it would still not be a priori clear that the resulting exponent is an integer.

To remedy this, we pass to a sufficiently large ring to contain everything, and arrive at a

diagram

Spec(Z((q4Nm)) ⊗ Z[1/(2Nm), ζ4Nm] //

Spec(Z((qw)) ⊗ Z[1/(2Nm), ζ4Nm]) // X(4N ; m)

Spec(Z((qw1m/δ)) ⊗ Z[1/(2Nm), ζ4Nm/δ ])φ

// Spec(Z((qw1)) ⊗ Z[1/(2N), ζ4N ]) // X1(4N)

where all maps (except, of course, φ) are the natural ones. Pulling back qw1 to the large ring on the

upper left gives φ(qw1) = ζlw1m/δq

δw1m/δ. As qw is a parameter on the analytic neighborhood of x

given by the original map 2.5, we must have

ζlw1m/δq

δw1m/δ = (unit) · qe

w

where e is the ramification index we seek. Now we simply solve for e.

2.7.1 The operators Tm, gcd(4N, m) = 1

In this section we take the base ring R to be a Z[1/(2Nm)]-algebra. A rational function Θ on

X(4N ; m) has divisor at least π∗1Σ4N,k − π∗

2Σ4N,k if and only if

(d∗Θ) ≥ d∗(π∗1Σ4N,k − π∗

2Σ4N,k) = π∗1Σ4N,k − π∗

2Σ4N,k


where d is the standard degeneracy map d : X1(4Nm)→ X(4N ; m) and πi = πi d.

Let us work out this divisor explicitly. Let (Tate(q), ζa4Nmqb

4Nm) be any level 4Nm structure

on Tate(q). Note that

π2(Tate(q), ζa4Nmqb

4Nm) = (Tate(q), ζa4N qb

4N ).

The other degeneracy map is somewhat more difficult to work out. First note that 〈ζamqb

m〉 = 〈ζlmqδ

m〉

for δ = gcd(b, m) and some l prime to δ (since any cyclic subgroup of order m is of this form for

some δ|m and l). We have

π1(Tate(q), ζa4Nmqb

4Nm) = (Tate(q)/〈ζlmqδ

m〉, ζa4N qb

4N/〈ζlmqδ

m〉).

By the proof of Lemma 10, this is equal to

(Tate(ζlm/δq

δm/δ), (ζ

a4N qb

4N )δ).

All this is defined over Z((q4Nm/δ))⊗ Z[1/(2Nm), ζ4Nm] and we will now rewrite it in these terms.

We have

(Tate(ζlδmqδ2

m ), ζδa4Nqδb

4N ) = (Tate(ζlδmq4Nδ

4Nm/δ), ζδa4N qmb

4Nm/δ)

= (Tate(ζlδmq4Nδ

4Nm/δ), ζaδ−bl4N (ζl

4Nm/δqδ4Nm/δ)

bm/δ)

which is in the “standard” form for Tate curves with level structure. Note that this arises from

(Tate(q), ζaδ−bl4N q

bm/δ4N ) via the change-of-rings

Z((q4N )) ⊗ Z[1/(2Nm), ζ4N ] −→ Z((q4Nm/δ))⊗ Z[1/(2Nm), ζ4Nm]

q4N 7−→ ζlδ4Nmqδ

4Nm/δ

Since m is odd, these two images of (the cusp associated to) (Tate(q), ζa4Nmqb

4Nm) lie above 1/2 if

and only if 2||b. In particular, the coefficient of this cusp on the divisor of interest is zero unless 2||b,

so let us assume this to be the case.

Let wi (i = 1, 2) denote the width of the image of the cusp associated to (Tate(q), ζa4Nmqb

4Nm)

under πi. By Lemma 10, the ramification index of π1 here is (δ2/m)(w/w1) and that of π2 is w/w2

(since it is easy to see that the ramification indices of d are simply the corresponding ratios of

widths). It follows that coefficient of our divisor at this cusp is zero unless 2||b, in which case it is

δ2

m

w

w1

kw1

4−

w

w2

kw2

4=

δ2 −m

4mkw. (2.6)

Theorem 11 Let k be an odd positive integer. There exists a rational function Θ on X(4N ; m)

with

(Θ) ≥ π∗1Σ4N,k − π∗

2Σ4N,k

if and only if m is a square, in which case Θ is unique and coincides with Θkm up to a constant

multiple.


Proof. If such a Θ exists then d∗Θ has divisor at least the one computed above on X1(4Nm). We

claim that this divisor is exactly the divisor of the pull-back of Θkm from X1(4m) (which we will

continue to call Θkm). The result will follow since (d∗Θ) ≥ (Θk

m) implies Θ = Θkm (up to a constant

multiple) and Θkm descends through d exactly when m is a square (here is where we use that k is

odd).

It was shown in section 2.6 that the order of vanishing of Θm at the cusp associated to the

level structure ζa4mqb

4m is zero unless 2||b in which case it is

δ2 −m

2δ,

since h = δ in this case. Thus the order of vanishing of the pull-back through X1(4Nm)→ X1(4m) at

the cusp associated to ζa4Nmqb

4Nm gets multiplied by the ramification index here, which is w/(2m/δ).

The result isδ2 −m

4mw.

Further multiplying by k to account for the power of k yields exactly the order (2.6) computed

above.

When m is a square, we define Tm to be the endomorphism of Mk/2(4N, χ, R) gotten by

taking Θ = Θm and C = mk/2−1 in (2.4).

Let us turn to the precise effect of Tp2 on the q-expansion at the cusp ∞, and show in

particular that our Tp2 agrees with Shimura’s (see [13]). Let k be an odd positive integer. Recall

that the cusp ∞ is represented by (Tate(q), ζ4N ). As in the above theorem, we break the sum

defining Tp2F into several pieces. We frequently refer without comment to results from section 2.6.

Let F ∈Mk/2(4N, χ, R) and let

F (Tate(q), ζ4N )

(∑

n∈Z

qn2

)k

be the q-expansion of F at ∞. First, we consider the lone subgroup 〈ζp2〉. We have

F (Tate(q)/〈ζp2〉, ζ4N )Θp2(Tate(q), ζ4, 〈ζp2〉)k = F (Tate(qp2

), ζp2

4N )

(∑n∈Z

qp2n2

∑n∈Z

qn2

)k

.

The contribution to the q-expansion is (see Definition 5)

F (Tate(qp2

), ζp2

4N )

(∑

n∈Z

qp2n2

)k

= χ(p2)∑

anqp2n.


Now consider the subgroups 〈ζip2qp2〉 for 0 ≤ i ≤ p2 − 1. We have

p2−1∑

i=0

F (Tate(q)/〈ζip2qp2〉)Θp2(Tate(q), ζ4, 〈ζ

ip2qp2〉)k =

=

p2−1∑

i=0

F (Tate(ζip2qp2), ζ4N )

(1

p

∑n∈Z

(ζip2qp2)n2

∑n∈Z

qn2

)k

.

The contribution to the q-expansion from these terms is

p−k

p2−1∑

i=0

∑an(ζi

p2qp2)n = p2−k∑

ap2nqn

where the sum is over those n divisible by p2.

Finally, we consider the contribution from the subgroups 〈ζjp2qp〉 where j runs through a

set of residues mod p which are prime to p. For convenience, we choose such a set of residues S

which are all 1 mod 4. We compute

∑

j∈S

F (Tate(q)/〈ζjp2qp〉, ζ4N )Θp2(Tate(q), ζ4, 〈ζ

jp2qp〉)

k =

=∑

j∈S

F (Tate(ζjpq), ζp

4N )

(1

εpp1/2

(j

p

) ∑n∈Z

(ζjpq)n2

∑n∈Z

qn2

)k

.

The contribution to the q-expansion is

ε−kp p−k/2χ(p)

∑

j∈S

(j

p

)∑an(ζj

pq)n = ε1−kp p(1−k)/2

∑(n

p

)anqn,

where we have used the formula

∑

j∈S

(j

n

)ζjnp =

(n

p

)∑

j∈S

(jn

p

)ζjnp =

(n

p

)εpp

1/2

for p 6 |n. The formula is clearly true for p|n as well. Note also that ε2p =

(−1p

).

Thus, multiplying the lot by pk−2 and putting it all together, we have the following theorem.

Theorem 12 Let F ∈ Mk/2(4N, χ, R) for an odd integer k. If∑

anqn is the q-expansion of F ,

then the q-expansion of Tp2F is∑

bnqn, where

bn = ap2n +

(−1

p

)(1−k)/2

p(k−1)/2−1

(n

p

)χ(p)an + pk−2χ(p2)an/p2 .


2.7.2 The operators Up2, p odd, p|N

The same procedure can be used to construct these operators. In fact, the same calculation shows

that the divisor of Θp2 on X(4N ; p2) is exactly π∗1Σ4N,k − π∗

2Σ4N,k. The only minor difference in

this case is that w1 and w2 are now generally different, while w1 = w2 in the previous case, though

this was never used.

2.7.3 The operators Up, p odd, p|N

In this section, we take out base ring to be any Z[1/(2Np)]-algebra. Define a function Θ on Y (4N ; p)

by

Θ(E, P, C) = Θp(E/C, (N/p)P/C)−1

where Θp is the unit on X1(4p) constructed in section 2.6. Let us first determine the divisor of Θ.

Let (ζa4N qb

4N , 〈ζlpq

δp〉) be a level structure on Tate(q). Performing the quotient and multi-

plication by N/p in the definition of Θ we get

(Tate(q)/〈ζlpq

δp〉, ζ

a4pq

b4p/〈ζ

lpq

δp〉) = (Tate(ζlδ

p q4pδ4p·p/δ), ζ

aδ−bl4p (ζl

4p·p/δqδ4p·/δ)

bp/δ).

By section 2.6, the q-order of Θp at this cusp is 0 unless 2||b in which case it is (h2−p)/4p.

Note that here we have h = gcd(bp/δ, p) = p/δ, since p|b implies vδ = 1. Thus the order of vanishing

of Θ at the cusp associated to our level structure is (assuming 2||b)

−wδ2

p

(p/δ)2 − p

4p= w

δ2 − p

4p

where w is the width of this cusp.

Next we must work out the coefficient of π∗1Σ4N,k − π∗

2Σ4N,k at this cusp. This proceeds

exactly as in previous cases and gives 0 unless 2||b, in which case it gives

δ2

p

w

w1

kw1

4−

w

w2

kw2

4= kw

δ2 − p

4p.

Thus (Θk) = π∗1Σ4N,k − π∗

2Σ4N,k, and thus, taking this Θ and C = p−1 in (2.4), we arrive at an

operator Up on the space of modular forms.

The caveat is that, although Up preserves the total space of modular forms, it does not

preserve the character. Indeed, since Θp has character (p/·), Up multiplies the character of its input

by (p/·)k = (p/·).

Let us work out the effect of Up on the q-expansion at ∞. Let F ∈Mk/2(4N, χ, R) and let

∑anqn = F (Tate(q), ζ4N )

(∑

n∈Z

qn2

)k


be the q expansion of F at ∞. The subgroups 〈ζlpqp〉 for l = 0, . . . , p− 1 are exactly the subgroups

of order p not meeting the subgroup generated by ζ4N . Note that

Θ(Tate(q), ζ4N , 〈ζlpqp〉) = Θp(Tate(q)/〈ζl

pqp〉, ζp/〈ζlpqp〉)

−1

= Θp(Tate(ζlpqp), ζp)

−1

=

(∑n∈Z

(ζlpqp)

pn2

∑n∈Z

(ζlpqp)n2

)−1

=

∑n∈Z

(ζlpqp)

n2

∑n∈Z

qn2 .

For a fixed l, the contribution to the sum defining UpF (Tate(q), ζ4N ) is

F (Tate(q)/〈ζlpqp〉, ζ4N/〈ζl

pqp〉)Θ(Tate(q), ζ4N , 〈ζlpqp〉)

k

= F (Tate(ζlpqp), ζ4N )

(∑n∈Z

(ζlpqp)

n2

∑n∈Z

qn2

)k

,

and the corresponding contribution to the q-expansion is

F (Tate(ζlpqp), ζ4N )

(∑

n∈Z

(ζlpqp)

n2

)k

=∑

an(ζlpqp)

n.

Summing over l and dividing by p gives the following result.

Theorem 13 There exists a unique map

Up : Mk/2(4N, χ, R) −→Mk/2(4N, χ(p/·)k, R)

having the effect∑

anqn 7−→∑

apnqn

on q-expansions at ∞.

2.7.4 The operator U4

Finally, we consider the operator U4. For this, we break with previous notation and introduce the

function

Θ4(τ) =θ(τ/4)

θ(τ).

This function is invariant under the group

Γ =

(a b

c d

)∈ SL2(Z)

∣∣∣∣∣ 4|b, 4|c

.


Let Y (4; 4) be the curve classifying isomorphism classes of triples (E, P, C) consisting of an elliptic

curve E, a point P of order 4, and a cyclic subgroup C of order 4 not meeting 〈P 〉. It has a compact-

ification X(4; 4) as usual which is defined over Z[1/2]. We have a complex analytic uniformization

Γ\h∼−→ Y (4; 4)an

τ 7−→

(Eτ ,

1

4,⟨τ

4

⟩),

which in turn gives a function Θ4 on Y (4; 4). This curve has 6 cusps represented by the elements of

∞, 0, 1, 2, 1/3, 1/2, all of which have width 4. The divisor of Θ4 is supported on those cusps with

either τ or τ/4 equivalent to 1/2 on X1(4), which are 1/2 and 2. A simple computation, similar to

that used to compute the divisor of Θm for m, odd shows that this divisor is in fact exactly

(Θ4) = [2]− [1/2].

By GAGA and the q-expansion principle, Θ4 yields an algebraic function on X(4; 4) defined over

Z[1/2]. The cusps 1/2 and 2 are represented by the level structures (ζ4q2, 〈q4〉) and (q4, 〈ζ4q2〉) on

Tate(q), respectively, and the divisor of Θ4 on X(4; 4)/Z[1/2] is again the difference (in the order

specified above) between these two.

We first compute π∗1Σ4N,k − π∗

2Σ4N,k. Let (ζa4N qb

4N , 〈ζl4q

δ4〉) be a level structure on Tate(q)

and let w be the width of the associated cusp and wi denote the width of the image of this cusp

under πi. The complicating issue in the U4 case is that it is no longer the case that one of these

images is equivalent to 1/2 if and only if the other is. We have

π1(Tate(q), ζa4Nqb

4N , 〈ζl4q

δ4〉) = (Tate(ζlδ

4 q4Nδ4N ·4/δ, ζ

aδ−bl4N (ζl

4N ·4/δqδ4N ·4/δ)

b·4/δ)

π2(Tate(q), ζa4Nqb

4N , 〈ζl4q

δ4〉) = (Tate(q), ζa

4N qb4N ).

The first of these is equivalent to 1/2 on X1(4) when 2||(4b/δ) and the second when 2||b. Note that

the condition

〈ζa4N qb

4N 〉 ∩ 〈ζl4q

δ4〉 = 0

implies that δ = 1 when b is even. As a result, the first of the above cusps is equivalent to 1/2

exactly when b is odd and δ = 2, and the second exactly when 2||b (and hence δ = 1).

Putting this together, we see that the coefficient of π∗1Σ4N,k−π∗

2Σ4N,k at the cusp associated

to (ζa4N qb

4N , 〈ζl4q

δ4〉) is

δ2

4ww1

kw1

4 b odd, δ = 2

− ww2

kw2

4 2||b, δ = 1

0 otherwise

=

kw4 b odd, δ = 2

−kw4 2||b, δ = 1

0 otherwise

.


Finally, we must compute the order of vanishing of the pull-back of Θ4 to X(4N ; 4) at this

cusp. The image of the cusp is 1/2 when 2||b (and hence δ = 1). The image is 2 when δ = 2 (and

hence b is odd). The ramification index is w/4, so the order of vanishing of the pull-back is

w

4·

1 b odd, δ = 2

−1 2||b, δ = 1

0 otherwise

.

Further multiplying by k, we arrive exactly at the divisor π∗1Σ4N,k − π∗

2Σ4N,k. Thus taking Θ = Θk4

and C = 1/4 in (2.4), we arrive at an endomorphism U4 of our space of modular forms.

Let us determine the effect of U4 on the q-expansion at ∞. Let F ∈ Mk/2(4N, χ, R) and

let∑

anqn = F (Tate(q), ζ4N )

(∑

n∈Z

qn2

)k

be its q-expansion at ∞. The cyclic subgroups of order 4 not meeting the subgroup generated by

ζ4N are the groups 〈ζl4q4〉 for l = 0, 1, 2, 3.

Note that the datum

(Tate(q), ζ4, 〈ζl4q4〉)

arises from that with l = 0 via the change-of-rings

Z((q4))⊗ Z[1/2] −→ Z((q4))⊗ Z[1/2, ζ4]

q4 7−→ ζl4q4.

It follows that

Θ4(Tate(q), ζ4, 〈ζl4q4〉) =

∑n∈Z

(ζl4q4)

n2

∑n∈Z

qn2 ,

since this is true for l = 0 by construction of Θ4. Thus the contribution to the sum defining

U4F (Tate(q), ζ4N ) corresponding to l is

F (Tate(q)/〈ζl4q4〉, ζ4N/〈ζl

4q4〉)Θ4(Tate(q), ζ4, 〈ζl4q4〉)

k =

= F (Tate(ζl4q4), ζ4N )

(∑n∈Z

(ζl4q4)

n2

∑n∈Z

qn2

)k

,

and the corresponding contribution to the q-expansion at ∞ is

F (Tate(ζl4q4), ζ4N )

(∑

n∈Z

(ζl4q4)

n2

)k

=∑

an(ζl4q4)

n.

Summing this over l and dividing by 4 we arrive at the following.


Theorem 14 There is a unique map

U4 : Mk/2(4N, χ, R) −→Mk/2(4N, χ, R)

having the effect∑

anqn 7−→∑

a4nqn

on q-expansions at ∞.

Remark 15 Provided 2|N , one can also construct an operator U2 which acts as∑

anqn 7→∑

a2nqn.

Predictably, this operator will alter the Nebentypus character of the form.

2.8 The operator W

In this section we assume that N is odd and define an Atkin-Lehner type operator W on

Mk/2(4N, χ, R). As usual, we fix a primitive fourth root of unity ζ4. Note that since N is odd, the

data of a point of order 4N is the same as that of a point of order 4 and a point of order N , the

correspondence being given by

P 7−→ (NP, 4P ).

In these terms, our map W is given by

(WF )(E, P, Q) = F (E/〈P 〉, P ′/〈P 〉, Q/〈P 〉)

where P ′ is chosen so that 〈P, P ′〉 = ζ4. Note that P ′ is not well-defined, but P ′/〈P 〉 is.

To verify that W preserves Mk/2(4N, χ, R) we need some facts about the Weil pairing on

Tate(q). The 4-torsion on the curve Tate(q)/R((q4N ))[ζ4] is rational over the base and sits in an exact

sequence

0→ µ4 → Tate(q)/R((q4N ))[ζ4][4]→ Z/4Z→ 0

where the second (nontrivial) map is given by q4 7→ 1. As a result, the Weil pairing on 4-torsion is

given determined by 〈ζ4, q4〉 = ζ4 (see [8]).

The (isomorphism classes of) level 4N structures on Tate(q) are of three kinds, according

to their image cusp on X1(4). In our current setting, they are the pairs (ζ4, P ), the pairs (ζ4q2, P ),

and the pairs (ζi4q4, P ) for i = 0, 1, 2, 3, where P is some point of order N . We check the order of

vanishing condition in each case separately.

In the first case, we note that 〈ζ4, q4〉 = ζ4, and compute

(WF )(Tate(q), ζ4, P ) = F (Tate(q)/〈ζ4〉, q4/〈ζ4〉, P/〈ζ4〉)

= F (Tate(q4), q, 4P ).


The cusp corresponding to this last level structure does not lie above 1/2, so this is holomorphic,

and hence WF is holomorphic at the cusp corresponding to (ζ4, P ).

In the second case, we have 〈ζ4q2, q4〉 = ζ4, so

(WF )(Tate(q), ζ4q2, P ) = F (Tate(q)/〈ζ4q2〉, q4/〈ζ4q2〉, P/〈ζ4q2〉)

= F (Tate(−q), q2, 2P ).

The triple F (Tate(−q), q2, 2P ) arises from the triple (Tate(q), ζ4q2, Q) (for an appropriate Q, speci-

fied below) via the change of rings

R((q2N ))[ζ4N ]φ−→ R((q2N ))[ζ4N ]

q2N 7−→ ζ4Nq2N ,

so the above expression is equal to

φ(F (Tate(q), ζ4q2, Q)).

Since φ does not alter the q-order of a Laurent series, this expression has order no worse

than that specified by the divisor Σ4N,k at the cusp corresponding to (ζ4, Q), which in turn depends

only on the width of the cusp. As we are trying to see that this whole expression has order no worse

than that specified by our divisor at the cusp corresponding to (ζ4q2, P ), is suffices to argue that

these two cusps have the same width. Suppose the P = ζaNqb

N , then Q = ζ2a−2bN q2b

N , and the result

is evident since N is odd.

Finally, noting that 〈ζi4q4, ζ

−14 〉 = ζ4, we compute

(WF )(Tate(q), ζi4q4, P ) = F (Tate(q)/〈ζi

4q4〉, ζ−14 /〈ζi

4q4〉, P/〈ζi4q4〉)

= F (Tate(ζi4q4), ζ

−14 , P ).

The cusp corresponding to this last level structure does not lie above 1/2, so this expression is

holomorphic.

The only remaining thing to check is that W preserves the character χ. This follows from

the fact that, for d ∈ (Z/4Z)× = ±1, we have 〈dP, dP ′〉 = 〈P, P ′〉. Thus

(WF )(E, dP, dQ) = F (E/〈dP 〉, dP ′, dQ)

= F (E/〈P 〉, dP ′, dQ)

= χ(d)F (E/〈P 〉, P ′, Q)

= χ(d)(WF )(E, P, Q).


2.9 Kohnen’s +-space

In [10] Kohnen introduces the subspace M+k/2(4) of the space of (holomorphic) modular

forms Mk/2(4) of weight k/2 and level 4 consisting of those forms whose q-expansions at ∞ are∑

anqn with an = 0 for (−1)(k−1)/2n ≡ 2, 3 (mod 4). He later generalizes this space to level 4N

with N odd and arbitrary character χ in [11]. In both papers, an operator on the space of modular

forms is introduced and it is proven that the +-space coincides with a particular eigenspace of this

operator. Moreover, Kohnen exploits the holomorphic setting in each of his proofs of this fact.

In this section, we generalize all of this to our setting of an arbitrary Z[1/2N ]-ring R and

character χ. In particular, we offer a completely algebraic proof that the +-space coincides with a

particular eigenspace of Kohnen’s operator. We continue to assume that N is odd for the remainder

of the section. As a result, our character χ factors as

χ(x) = χ4(x)χN (x)

for a uniquely defined pair of characters χ4 and χN modulo 4 and N , respectively. Let ε = χ4(−1)(=

χN (−1)).

Let M+k/2(4N, χ, R) denote the subspace of Mk/2(4N, χ, R) whose q-expansions at ∞ are

of the form∑

anqn with an = 0 unless ε(−1)(k−1)/2n ≡ 0, 1 (mod 4). Following Kohnen, we define

an operator K on Mk/2(4N, χ, R) by K = W U4. Let

λ0 =χn(4)

2(ε(1− ζ4)

k + (1 + ζ4)k).

Theorem 16 The space M+k/2(4N, χ, R) is exactly the λ0-eigenspace of K.

Remark 17 Note that many of our conventions differ from those of Kohnen. In particular, k means

something different for Kohnen and the operator K is normalized differently in [11], where it is called

Q.

We first prove that the plus space is contained in the λ0-eigenspace of K. Our proof is a

direct adaptation of that given in [11].

Fix F ∈Mk/2(4N, χ, R) and let∑

anqn denote it’s q-expansion at ∞. We define a map

K ′ : Mk/2(4N, χ, R)→Mk/2(8N, χ, R)

as follows. For a triple (E, P, Q) with P of order 8 and Q of order N , we first choose a point P ′ of

order 4 (well-defined up to 〈2P 〉) with 〈2P, P ′〉 = ζ4. Let E′ = E/〈2P 〉, and note that the cyclic

subgroups of order 4 in E′ not meeting that generated by P ′ now come in two flavors, namely, those

which contain the image of P on E′ and those which do not. We prefer the latter, and sum over


this set in the following definition.

(K ′F )(E, P, Q) :=1

4

∑

P/〈2P 〉/∈C

F (E′/C, P ′/C, Q/C)Θ4(E′, P ′, C)k

We could, of course, sum over the set which do contain the image of P on E′ and thereby get some

other operator K ′′. Of course, it would follow immediately that K = K ′ +K ′′. That is, K ′ is sort of

“half” of K. We single it out because it is the nice half in the sense that it has a predictable effect

on the q-expansion at infinity.

Interestingly, there is another way to recover K from K ′.

Proposition 18 Let Tr denote the trace map from Mk/2(8N, χ, R) to Mk/2(4N, χ, R). Then we

have KF = 12Tr(K ′F ).

Proof. Let E′ and P ′ be as above.

Tr(K ′F )(E, R, Q) =∑

2P=R

(K ′F )(E, P, Q)

=∑

2P=R

∑

P/〈R〉/∈C

F (E′/C, P ′/C, Q/C)Θ4(E′, P ′, C)k

The assertion follows from this and the fact that, for any C in E′ not meeting 〈P ′〉, there are exactly

two points P with 2P = R such that C does not contain the image of P .

We turn to the effect on the q-expansion at infinity. For the triple (Tate(q), ζ8, ζN ), we

have and E′ = Tate(q)/〈ζ4〉 = Tate(q4). Also, P ′ equals the image of q4, or q, and the image of ζ8

here is −1. The set of cyclic subgroups of E′ of order 4 not meeting 〈q〉 and not containing −1 is

〈ζ4q〉, 〈ζ−14 q〉. Thus

4(K ′F )(Tate(q), ζ8, ζN )

= F (Tate(q4)/〈ζ4q〉, q/〈ζ4q〉, ζ4N/〈ζ4q〉)Θ4(Tate(q4), q, 〈ζ4q〉)

k

+ F (Tate(q4)/〈ζ−14 q〉, q/〈ζ−1

4 q〉, ζ4N/〈ζ−1

4 q〉)Θ4(Tate(q4), q, 〈ζ−14 q〉)k

= χN (4)F (Tate(ζ4q), q, ζN )Θ4(Tate(q4), q, 〈ζ4q〉)k

+ χN (4)F (Tate(ζ−14 q), q, ζN )Θ4(Tate(q4), q, 〈ζ−1

4 q〉)k

= εχN (4)F (Tate(ζ4q), ζ4, ζN )Θ4(Tate(q4), q, 〈ζ4q〉)k

+ χN (4)F (Tate(ζ−14 q), ζ4, ζN )Θ4(Tate(q4), q, 〈ζ−1

4 q〉)k

(2.7)

where we have used the relation q = ζ−14 on Tate(ζ4q) and the relation q = ζ4 on Tate(ζ−1

4 q).

The next step is to compute the Θ4’s in the last expression explicitly. To accomplish this,

we let j = ±1 and note that the matrix(

1 j

−j 0

)


furnishes an isomorphism

(E4τ , τ,

⟨j

4+ τ

⟩)∼−→

(E 4τ+j

−4jτ,1

4,

⟨4τ+j−4jτ

4

⟩)

z 7−→z

4τ

(this is the usual isomorphism possibly followed by inversion on the elliptic curve, depending on the

sign of j). So the holomorphic q-expansion of Θ4(Tate(q4), q, 〈ijq〉) is given by

θ(

14

4τ+j−4jτ

)

θ(

4τ+j−4jτ

) =θ(

(τ+j/4)−4j(τ+j/4)+1

)

θ(−1− 1

4τ

)

=(−4jτ)1/2θ(τ + j/4)

(−i2τ)1/2θ(τ)

= (1− ji)

∑n∈Z

ijn2

qn2

∑n∈Z

qn2 ,

where we once again have been careful about the choice of square root. Thus our algebraic q-

expansion is

(1− jζ4)

∑n∈Z

ζjn2

4 qn2

∑n∈Z

qn2 . (2.8)

It follows from this and (2.7) that K ′ has the following effect on q-expansions

(K ′F )(Tate(q), ζ8, ζN )

(∑

n∈Z

qn2

)k

=χN (4)

4

∑

n

[ε(1− ζ4)kζn

4 + (1 + ζ4)kζ−n

4 ]anqn.

An easy computation shows that

ε(1− ζ4)kζn

4 + (1 + ζ4)kζ−n

4 = (ε(1− ζ4)k + (1 + ζ4)

k) ·

1 ε(−1)

k−12 n ≡ 0, 1 (mod 4)

−1 ε(−1)k−12 n ≡ 2, 3 (mod 4)

.

Finally, suppose that F ∈M+k/2(4N, χ, R). Then by the q-expansion principle

K ′F = χN (4)(ε(1 − ζ4)k + (1 + ζ4)

k)F.

Taking the trace of both sides and applying Proposition 18 we arrive at

KF =1

2Tr(K ′F ) =

χN(4)

8(ε(1− ζ4)

k + (1 + ζ4)k)Tr(F ) = λ0F.

The proof that the λ0-eigenspace of K is contained in the plus space is somewhat more

involved. It is here that Kohnen makes essential use of the holomorphic setting in his analyses. We

get around this by offering a completely different, and totally algebraic, argument. We begin by

writing out the all of (KF )(Tate(q), ζ4, ζN ), the “nice half” of which occurs above.


(KF )(Tate(q), ζ4, ζN ) = (U4F )(Tate(q)/〈ζ4〉, q4/〈ζ4〉, ζN/〈ζ4〉)

= (U4F )(Tate(q4), q, ζ4N )

Note that the set of cyclic subgroups of order 4 not meeting that generated by q is

〈ζ4〉, 〈ζ4q2〉, 〈ζ4q〉, 〈ζ

−14 q〉,

so the above becomes

1

4(F (Tate(q4)/〈ζ4〉, q/〈ζ4〉, ζ

4N/〈ζ4〉)Θ4(Tate(q4), q, 〈ζ4〉)

k

+ F (Tate(q4)/〈ζ4q2〉, q/〈ζ4q

2〉, ζ4N/〈ζ4q

2〉)Θ4(Tate(q4), q, 〈ζ4q2〉)k

+ F (Tate(q4)/〈ζ4q〉, q/〈ζ4q〉, ζ4N/〈ζ4q〉)Θ4(Tate(q4), q, 〈ζ4q〉)

k

+ F (Tate(q4)/〈ζ−14 q〉, q/〈ζ−1

4 q〉, ζ4N/〈ζ−1

4 q〉)Θ4(Tate(q4), q, 〈ζ−14 q〉)k

=1

4(F (Tate(q16), q4, ζ16

N )Θ4(Tate(q4), q, 〈ζ4〉)k

+ F (Tate(−q4), q2, ζ8N )Θ4(Tate(q4), q, 〈ζ4q

2〉)k

+ F (Tate(ζ4q), q, ζ4N )Θ4(Tate(q4), q, 〈ζ4q〉)

k

+ F (Tate(ζ−14 q), q, ζ4

N )Θ4(Tate(q4), q, 〈ζ−14 q〉)k)

To simplify this somewhat further, we again exploit that the relations q = ζ−14 and q = ζ4

hold on the third and fourth line, respectively. Using this and the nebentypus character, we arrive

at

4(KF )(Tate(q), ζ4, ζN ) = χN (16)F (Tate(q16), q4, ζN )Θ4(Tate(q4), q, 〈ζ4〉)k

+ χN (8)F (Tate(−q4), q2, ζN )Θ4(Tate(q4), q, 〈ζ4q2〉)k

+ εχN(4)F (Tate(ζ4q), ζ4, ζN )Θ4(Tate(q4), q, 〈ζ4q〉)k

+ χN (4)F (Tate(ζ−14 q), ζ4, ζN )Θ4(Tate(q4), q, 〈ζ−1

4 q〉)k.

(2.9)

We can compute the q-expansions of the various Θ4’s as usual by passing to the analytic

world (see Section 1.3). We need only work out the first two, as the remaining two have already

been done (they are given by (2.8)). For the first term, the matrix

(0 −1

1 0

)

gives the isomorphism(

E4τ , τ,

⟨1

4

⟩)∼−→

(E−1

4τ,1

4,

⟨ −14τ

4

⟩)

z 7−→z

4τ.


The analytic q-expansion of Θ4(Tate(q4), q, 〈i〉) is then

θ(

14−14τ

)

θ(−14τ

) =ϑ(−18τ

)

ϑ(−12τ

)

=(−i8τ)1/2ϑ(8τ)

(−i2τ)1/2ϑ(2τ)

= 2

∑n∈Z

q4n2

∑n∈Z

qn2 ,

where we have been careful to pick the correct square roots.

For the second term, we use the isomorphism

(E4τ , τ,

⟨1

4+ 2τ

⟩)∼−→

(E 8τ−1

4τ,1

4,

⟨ 8τ−14τ

4

⟩)

z 7−→z

4τ.

The analytic q-expansion of Θ4(Tate(q4), q, 〈iq2〉) is then

θ(

14

8τ−14τ

)

θ(

8τ−14τ

) =ϑ(1− 1

8τ

)

θ(2− 1

4τ

)

=(−i8τ)1/2q

∑n∈Z

q4(n2+n)

(−i2τ)1/2∑

n∈Zqn2

= 2q∑

n∈Zq4(n2+n)

∑n∈Z

qn2

We now apply the formula (2.9) with F replaced by κF . We compute the new terms

involving K that turn up in the first two (but not the last two!) terms.

(KF )(Tate(q16), q4, ζN ) = (U4F )(Tate(q16)/〈q4〉, ζ−14 /〈q4〉, ζN/〈q4〉)

= ε(U4F )(Tate(q4), ζ4, ζN )

As the collection of cyclic subgroups of order four not meeting that generated by ζ4 is 〈ζi4q〉i=0,1,2,3,


this is equal to

ε

4

3∑

i=0

F (Tate(q4)/〈ζi4q〉, ζ4/〈ζ

i4q〉, ζN/〈ζi

4q〉)Θ4(Tate(q4), ζ4, 〈ζi4q〉)

k

=ε

4

3∑

i=0

F (Tate(ζi4q), ζ4, ζN )Θ4(Tate(q4), ζ4, 〈ζ

i4q〉)

k

=ε

4

3∑

i=0

φi

(F (Tate(q), ζ4, ζN )Θ4(q

4, ζ4, 〈q〉)k)

=ε

4

3∑

i=0

φi

F (Tate(q), ζ4, ζN )

( ∑n∈Z

qn2

∑n∈Z

q4n2

)k

=ε

4(∑

n∈Zq4n2

)k3∑

i=0

φi

(∑

n

anqn

)

=ε

(∑n∈Z

q4n2)k∑

n

a4nq4n,

where φi is the change of rings given by q 7→ ζi4q.

To compute (KF )(Tate(−q4), q2, ζN ) we must find a point P on Tate(−q4) with 〈q2, P 〉 =

ζ4. Consider the curve Tate(q) over the base Z((q4))⊗R[ζ4] (it is, of course, defined over a smaller

ring). Pick a primitive eight root ζ8 in the ring Z((q)) ⊗R[ζ8]. Then, under the change of rings

Z((q4))⊗R[ζ4] −→ Z((q)) ⊗R[ζ8]

q4 7−→ ζ8q,

Tate(q) becomes Tate(−q4) (also defined over a smaller ring). The point of these larger rings is that

the 4-torsion is now rational, and we may use the fact that 〈ζ−14 q2, q

−14 〉 = ζ4 on Tate(q) and the

compatibility of the Weil pairing under change of ring (see [8]) to conclude that 〈q2, ζ−18 q−1〉 = ζ4.

Thus,

(KF )(Tate(−q4), q2, ζN ) = (U4F )(Tate(−q4)/〈q2〉, ζ−18 q−1/〈q2〉, ζN/〈q2〉)

= (U4F )(Tate(q4), ζ−14 q−2, ζ2

N )

= εχN(2)(U4F )(Tate(q4), ζ4q2, ζN ).

The cyclic subgroups of order four not meeting that generated by ζ4q2 are 〈ζi

4q〉, i = 0, 1, 2, 3, so,


continuing we get

εχN (2)

4

3∑

i=0

F (Tate(q4)/〈ζi4q〉, ζ4q

2/〈ζi4q〉, ζN/〈ζi

4q〉)Θ4(Tate(q4), ζ4q2, 〈ζi

4q〉)k

=εχN (2)

4

3∑

i=0

F (Tate(ζi4q), ζ4q

2, ζN )Θ4(Tate(q4), ζ4q2, 〈ζi

4q〉)k

=εχN (2)

4

3∑

i=0

φi

(F (Tate(q), (−1)iζ4q

2, ζN )Θ4(Tate(q4), (−1)iζ4q2, 〈q〉)k

)

where φ is the change of rings given by replacing q by ζi4q. Note that (ζ4q

2)−1 = −ζ4q2 on the curve

Tate(q4), so the isomorphism class of the data fed into Θ4 is independent of i. Also, the relation

(−1)iζ4q2 = ζ

(−1)i

4 holds on Tate(q), so the above becomes

εχN (2)

4

3∑

i=0

εiφi(F (Tate(q), ζ4, ζN )Θ4(Tate(q4), ζ4q2, 〈q〉)k).

The matrix (1 0

2 1

)

gives the isomorphism

(E4τ ,

1 + 8τ

4, 〈τ〉

)∼−→

(E 4τ

8τ+1,1

4,

⟨τ

8τ + 1

⟩)

z 7−→z

8τ + 1

shows that the analytic q-expansion of the Θ4 occurring in the last expression above is

θ(

14

4τ8τ+1

)

θ(

4τ8τ+1

) =(8τ + 1)1/2θ(τ)

ϑ(1− 1

8τ+1

)

=(8τ + 1)1/2

∑n∈Z

qn2

(−i(8τ + 1))1/2ζ8q∑

n∈Zq4(n2+n)

=

∑n∈Z

qn2

q∑

n∈Zq4(n2+n)

where we have used transformation formulas for theta functions with characteristics, being careful

to choose the correct square roots.


Substituting this,

εχN(2)

4

3∑

i=0

εiφi

F (Tate(q), ζ4, ζN )

( ∑n∈Z

qn2

q∑

n∈Zq4(n2+n)

)k

=εχN (2)

4

3∑

i=0

εiφi

( ∑n anqn

(q∑

n∈Zq4(n2+n))k

)

=εχN (2)

4(q∑

n∈Zq4(n2+n))k

3∑

i=0

∑

n

εiζi(n−k)4 anqn.

Now note that

3∑

i=0

εiζi(n−k)4 =

4 ε(−1)

k−12 n ≡ 1 (mod 4)

0 otherwise,

so the above isεχN(2)

(q∑

n∈Zq4(n2+n))k

∑

n∈S

anqn,

where S denotes the set of n with ε(−1)(k−1)/2n ≡ 1 (mod 4).

Substituting all this into the equation (2.9), with F replaced by KF , we get

4(K2F )(Tate(q), ζ4, ζN )

= χN (16)

(ε

(∑

n∈Zq4n2)k

∑

n

a4nq4n

)(2

∑n∈Z

q4n2

∑n∈Z

qn2

)k

+χN(8)

(εχN (2)

(q∑

n∈Zq4(n2+n))k

∑

n∈S

anqn

)(2q∑

n∈Zq4(n2+n)

∑n∈Z

qn2

)k

+εχN(4)(KF )(Tate(ζ4q), ζ4, ζN )

((1− ζ4)

∑n∈Z

(ζ4q)n2

∑n∈Z

qn2

)k

+χN(4)(KF )(Tate(ζ−14 q), ζ4, ζN )

((1 + ζ4)

∑n∈Z

(ζ−14 q)n2

∑n∈Z

qn2

)k

.

Now assume that F is an eigenform for K with some eigenvalue λ. After multiplying by

(∑

n∈Zqn2

)k, this equation becomes

4λ2∑

n

anqn = εχN(16)2k∑

n

a4nq4n + εχN(16)2k∑

n∈S

anqn

+χN(4)λ∑

n

[ε(1− ζ4)kζn

4 + (1 + ζ4)kζ−n

4 ]anqn.

Examining the terms in this equation corresponding to those n with ε(−1)(k−1)/2n ≡ 0, 1 (mod 4)

(i.e. both 0 and 1 give the same equation) we arrive at the equation

(4λ2 − χN (4)[ε(1− ζ4)k + (1 + ζ4)

k]λ− εχN (16)2k)an = 0,


while the remaining terms (both) give the equation

(4λ2an + [ε(1− ζ4)k + (1 + ζ4)

k]λ)an = 0.

One easily checks that λ0 is a root of the first polynomial and not a root of the second.

Thus, assuming that F is an eigenform with eigenvalue λ0, exactly those an with ε(−1)(k−1)/2n ≡ 2, 3

(mod 4) are forced to vanish. This completes the proof of the theorem.

Chapter 3

p-adic Modular Forms of

Half-Integral Weight

3.1 Overconvergent p-adic Modular Forms of Half-Integral

Weight

Let p ≥ 5 be a rational prime, let R be a p-adic ring, that is, one for which the natural

map

R −→ lim←−

R/pnR

is an isomorphism, and let r ∈ R. In this section, we define the space Mk/2(4N, R, r) of r-

overconvergent p-adic modular forms of weight k/2 and level 4N . Given the formalism developed in

the first part of this paper, the definition mimics that in [7] in the integral weight case.

Recall that, if p = 0 in R, then there is a canonically defined modular form A called

the Hasse invariant, which is of weight p − 1 and level 1. To define A, we must specify the value

A(E/S , ω) for each elliptic curve E/S (S and R-algebra) and nonvanishing differential ω on E.

Serre-Grothendieck duality determines a dual base η of H1(E, OE) and we define A(E, ω) by

Frob∗η = A(E, ω)η,

where Frob is the Frobenius endomorphism of OE . Note that if λ ∈ S×, then λω determines the

dual base λ−1η and

Frob∗(λ−1η) = λ−pFrob∗η = λ−pA(E, ω)η = λ1−pA(E, ω)(λ−1η).

Thus

A(E, λω) = λ1−pA(E, ω),

41

Chapter 3: p-adic Modular Forms of Half-Integral Weight 42

and A is therefore a (possibly non-holomorphic at∞) modular form of weight p− 1. It is a theorem

of Deligne that the q-expansion of A is the constant 1 (see [7]). In particular, A is holomorphic and

defined over Fp, by the q-expansion principle.

Let N ≥ 5 so that we have a fine moduli scheme X1(N) and bundle ω. Then A furnishes

a section of ωp−1 on X1(N)/R. As such, A vanishes simply at each supersingular elliptic curve, and

nowhere else. Since p− 1 is even and ω2 makes good sense on the coarse moduli scheme X1(4), the

above works in level 4 also.

Since p ≥ 5, the usual Eisenstein series of weight p− 1 and level 1, namely

Ep−1(τ) = 1−2(p− 1)

2bp−1

∞∑

n=1

σp−2(n)qn (3.1)

has coefficients in Q∩Zp. In particular, by the q-expansion principle, Ep−1 is defined over this ring,

and can accordingly be reduced mod p to obtain a modular form over Fp. The q-expansion above

is well-known to be congruent to 1 mod p ([7]), so by the q-expansion principle, we see that the

reduction of Ep−1 mod p is precisely A. If p = 2 or 3, then there is no such lifting of A in level 1

([7]).

Katz exploits this lifting Ep−1 in an ingenious way to define overconvergent modular forms

in integral weight, where one wants to limit one’s moduli problem to elliptic curves which are “not

too supersingular” in the sense that the valuation of Ep−1 is not too large.

We take as test data triples (E, P, Y ) with π : E → S an elliptic curve over a p-adically

complete R-algebra S, P a point of order 4N , and Y a section of ω1−pE/S , where ωE/S = π∗ΩE/S ,

satisfying Y · Ep−1(E) = r. We will refer to such a thing (perhaps without the level structure P ),

as an r-test datum if we wish to specify the growth condition.

Definition 19 An r-overconvergent p-adic modular form of weight k/2 (k odd) and level 4N is a

rule which assigns to each test datum as above, an element of S, subject to the following conditions.

Firstly, the rule must be insensitive to isomorphisms (over S) of the data involved and compatible

with arbitrary pull-back under maps of complete p-adic R-algebras. Secondly, for any point P on

Tate(q), we require

F (Tate(q), P, r(Ep−1(Tate(q)))−1) ∈ q−e4NR[[q4N ]],

where

e =4N

word(Tate(q),P )Σ4N,k.

The R-module of such forms in denoted Mk/2(4N, R, r).

Remark 20 The divisor Σ4N,k here is taken on the curve X1(4N)/R. Also, the expression (Ep−1(Tate(q)))−1

makes sense, as, by the above remarks,

Ep−1(Tate(q)) = E(q)ωp−1can


where ωcan is the canonical nonvanishing differential on Tate(q) (see [7]) and E(q) is the q-expansion

(3.1), which is congruent to 1 mod p, and hence invertible.

For a character χ : (Z/(4N)Z)× → C×, we get the space Mk/2(4N, χ, R, r) of forms with

Nebentypus by adding the condition F (E, dP, Y ) = χ(d)F (E, P, Y ) on our rule.

For later use, we will also introduce a version of the above definition where we place no

conditions at the cusps whatsoever. That is, we remove the condition on the Tate curve in the

definition above. The space of such forms will be denoted F (4N, R, r).

Since we have assumed R to be complete, it suffices in the above definition to work only

with test data over R-algebras S in which p is nilpotent. As a formal consequence one sees that

Mk/2(4N, R, r) = lim←−n

Mk/2(4N, R/pnR, r)

and similarly for F (4N, R, r). In particular, if R is a mixed characteristic DVR of residue charac-

teristic p and K is its field of fractions, this observation implies that if we place a norm of the space

Mk/2(4N, R, r)⊗K (or F (4N, R, r)⊗K) by declaring Mk/2(4N, R, r) (respectively F (4N, R, r)) to

be the closed unit ball, then these spaces become Banach spaces over K.

3.2 The situation when p is nilpotent

Suppose that p is nilpotent in R. Then the words “p-adically complete” can be removed

and the situation is now completely algebraic. The upshot of this is that there is now a moduli space

of r-test data.

Following Katz ([7]) we construct this moduli space as follows. Assume N ≥ 4. Starting

with the space Y1(N)/R and the sheaf L = ω1−p, we note that the the space

SymmY1(N)L −→ Y1(N)

is the moduli space parameterizing triples (E/S , P, Y ) where E/S is an elliptic curve over an R-

algebra S, P is a point of order N , and Y is any section of ω1−pE/S . The section Ep−1 of ωp−1

Y1(N) = L

gives a regular function on this moduli space, and we define Y1(N)(r)/R to be the closed subscheme

defined by the vanishing of Ep−1 − r.

Y1(N)(r) // SymmY1(N)L

Y1(N)

Then Y (N)(r)/R parameterizes actual r-test data (E, Y, P ).


One can similarly construct a version X1(N)(r) where the cusps have been added (provided

N ≥ 5) and versions Y (N ; m)(r) and X(N ; m)(r) by adding a cyclic subgroup of order m not meeting

that generated by P to the level structure.

For a general p-adic ring R, one obtains in this manner a compatible system of spaces

Y1(N)(r)n = Y1(N)(r)/(R/pnR) in the sense that Y1(N)(r)n is the reduction mod pn of the space

Y1(N)(r)n+1 for each n ≥ 1.

3.3 Good p-adic Hecke Operators

Fix l a rational prime not dividing 2Np. In this section we wish to define the operators

Tl2 on Mk/2(4N, R, r). In the case of integral weight, this construction is given in [7] and [5]. In

both articles the construction is incorrect (for the same reason). The problem that arises for these

authors is described and corrected below.

Looking at the definition of Tl2 in the non-p-adic case, one is led to consider, for a par-

ticular test object (E, P, Y ) over (the p-adically complete R-algebra) S, the set of test objects

(E/C, P/C, YC) where C runs over the set of cyclic subgroups of order l2, and YC is chosen to

satisfy

Ep−1(E/C) · YC = r. (3.2)

It is in the choice of YC that the above authors run into problems. Indeed, letting π denote the

natural map E → E/C, the section YC = π∗Y is used in [7] and [5]. The problem is that this does

not, in general, satisfy the condition (3.2). As the following proposition demonstrates, it does satisfy

(3.2) in the case p = 0 in S.

Proposition 21 Let (E, P, Y ) be an r-test datum over an Fp-algebra S and let C be a cyclic sub-

group of order m (prime to 2Np). Let π denote the natural map E → E/C. Then the triple

(E/C, P/C, π∗Y ) is also an r-test datum.

Proof. Since Ep−1 lifts the Hasse invariant A, we can replace Ep−1 by A in (3.2) and use the

definition given in Section 3.1. Let ω be a nonvanishing section of ΩE/S and let η be its dual in

H1(E, OE). Since p does not divide m, π is etale, so that π∗ω is nonvanishing on E/C. By naturality

of the Serre-Grothendieck duality pairing, π∗η is dual to π∗ω. By definition, we have

Frob∗η = A(E, ω)η.

Thus

Frob∗π∗η = π∗Frob∗η = A(E, ω)π∗η,

so that, by definition, we have

A(E/C, π∗ω) = A(E, ω).


Translating this into the language of sections of ωp−1 gives

π∗A(E) = A(E/C).

Now since Y ·A(E) = r it follows immediately that π∗Y ·A(E/C) = r as well.

Fix a p-adic ring R which is flat over Zp. Assume that, in R, r|p and p/r is topologically

nilpotent. For example, we may take for R a complete DVR of generic characteristic 0 and residue

characteristic p in which v(r) < v(p). We now wish to construct a correspondence which associates

to any r-test datum (E, P, Y ) over a p-adically complete R-algebra S and any cyclic subgroup C of

order m of E, a section YC of ω1−p(E/C)/S such that (E/C, P/C, YC) is an r-test datum. Moreover,

we want this correspondence to be compatible in the obvious sense with isomorphism (over S) of

the data involved and compatible under pull-back through arbitrary change of p-adically complete

R-algebra S.

Let A(N, m) be the coordinate ring of the (affine) modular scheme Y (N ; m)/R and let

(E, P, C)/A(N ;m) be the associated universal triple. By the proof of Proposition 21, there exists a

section α of ωp−1(E/C)/A(N,m) such that

Ep−1(E/C) = π∗Ep−1(E) + pα.

Moreover, by our hypotheses on R, this α is unique.

Let S, (E, P, Y ), C, and π be as above. The data (E, P, C) furnishes a unique pair of maps

(φ, Φ) such that the diagram

E

Φ// E

Spec(S)φ

// Spec(A(N ; m))

is Cartesian and compatible with all of the data P, C, P, C. Moreover, the map φ depends only on

the isomorphism class of (E, P, C). We define a section YC of ω1−p(E/C)/S as follows

YC =π∗Y

1 + (p/r)Φ∗(α)π∗Y.

This makes sense by completeness of S since p/r is topologically nilpotent.

Proposition 22 As defined above, (E/C, P/C, YC) is an r-test datum. This associated is compatible

with isomorphisms of the data involved and compatible with pullbacks under changes of (p-adically

complete) R-algebras S.

Proof. To verify the first claim, we simply compute. Let π : E → E/C and Π : E→ E/C denote the


natural maps.

YC ·Ep−1(E/C) =π∗Y

1 + (p/r)Φ∗α · π∗Y·Φ∗(Ep−1(E/C))

=π∗Y

1 + (p/r)Φ∗α · π∗Y·Φ∗(Π∗Ep−1(E) + pα)

=π∗Y · π∗Ep−1(E) + pΦ∗α · π∗Y

1 + (p/r)Φ∗α · π∗Y= r

Suppose that (E′, P ′, C′, Y ′)/S is an isomorphic quadruple given by the S-isomorphism

F : E → E′. By uniqueness of the pair (φ, Φ), there is a commutative diagram

E

##GGG

GGGG

GGF

//

Φ

$$E′

Φ′

// E

Spec(S)φ

// Spec(A(N ; m))

Thus

π∗Y

1 + (p/r)Φ∗Eα · π∗Y

=π∗F ∗Y ′

1 + (p/r)F ∗Φ′∗α · π∗F ∗Y ′

=F ∗π′∗Y ′

1 + (p/r)F ∗Φ′∗α · F ∗π′∗Y ′

=F ∗π′∗Y ′

1 + (p/r)F ∗(Φ′∗α · π′∗Y ′).

Now Φ′∗α · π′∗Y ′ ∈ S and F is an S-morphism, so the F ∗ fixes this, and the above reduces to

F ∗

(π′∗Y ′

1 + (p/r)Φ′∗α · π′∗Y ′

),

as desired.

Now let f : S → S′ be a map of p-adically complete R-algebras and let (E, P, Y ) and C be

as above. Denote by (E′, P ′, Y ′) and C′ the base change of these objects to S′ (not to be confused

with the isomorphic objects of the previous paragraph). Then, again by uniqueness, we have a

commutative diagram

E′

F//

ΦE′

%%E

ΦE// E

Spec(S′)f

// Spec(S)φ

// Spec(A(N ; m))


We compute

π′∗Y ′

1 + (p/r)Φ′∗α · π′∗Y ′=

π′∗F ∗Y

1 + (p/r)F ∗Φ∗Eα · π′∗F ∗Y

=F ∗π∗Y

1 + (p/r)F ∗(Φ∗Eα · π∗Y )

= F ∗

(π∗Y

1 + (p/r)ΦEα · π∗Y

),

which is the compatibility we seek.

Recall that l is a rational prime not dividing 2Np. Let R be a Zp-flat p-adic ring in which

r|p and p/r is topologically nilpotent. Let F ∈Mk/2(4N, R, r). We define a rule Tl2F by

(Tl2F )(E, P, Y ) = lk−2∑

C

F (E/C, P/C, YC)Θl2(E, P, C)k

where the sum is over the set of cyclic subgroups of E of order l2 and YC is as constructed above.

By Proposition 22 this rule satisfies the compatibilities of Definition 19.

We must check that Tl2F satisfies the order of vanishing requirements imposed by Σ4N,k.

By completeness and passage to the limit, it suffices to check this modulo pn for n ≥ 1. Recall from

Section 3.2 that we have compatible families of moduli spaces X1(4N)(r)n and X(4N ; m)(r)n. By

Proposition 22 we have constructed compatible families of correspondences

Y (4N ; m)(r)n

π1

yysssssssssπ2

%%KKKKKKKKK

Y1(4N)(r)n Y1(4N)(r)n

where π2 forgets the cyclic subgroup of order m and π1 quotients by it in accordance with the above

construction.

We wish to compactify these correspondences. Over the ordinary locus of the above curves

the quantity section Ep−1 is invertible. It follows that the “Y ” in the r-test data is uniquely

determined on this locus, and one has a canonical identification with the ordinary parts of these

curves and the ordinary parts of the modular curves Y (4N ; m) and Y1(4N). Using this, on can glue

the above correspondences with the usual ones on X1(4N) along the ordinary locus to compactify

the above correspondences, and arrive at a compatible family of correspondences

X(4N ; m)(r)n

π1

yyssssssss

ssπ2

%%KKKK

KKKKKK

X1(4N)(r)n X1(4N)(r)n


By construction, all of these spaces admit a map which “forgets Y ,” and all fit into a

compatible collection of diagrams as follows.

X(4N ; m)(r)n

π1

yyssssss

ssss

π2

<<

<<<<

<<<<

<<<<

<<<<

// X(4N ; m)n

π1

xxxxxxxx

π2

7

77

77

77

77

77

77

77

77

X1(4N)(r)n// X1(4N)n

X1(4N)(r)n// X1(4N)n

Pulling back the equality

(Θl2) = π∗1Σ4N,k − π∗

2Σ4N,k

of section 2.7 through these forgetful maps gives us the same equality of divisors on X(4N ; m)(r)n.

Thus, modulo pn, our p-adic Hecke operator Tl2 becomes the familiar

H0(X1(4N)(r)n, Σ4N,k) 7−→ H0(X1(4N)(r)n, Σ4N,k)

F 7−→ lk−2π2∗(π∗1F ·Θl2)

and in particular preserves the divisor Σ4N,k.

Finally, we address the effect of Tl2 on the q-expansion at ∞. Since Ep−1(Tate(q)/C) is

invertible for any subgroup C of Tate(q), the Y making (E/C, Y ) an r-test datum is unique, and we

need not appeal to the above construction of YC . The calculation of the q-expansion of Tl2F follows

that in Section 2.7 verbatim - one simply has to carry around r(Ep−1(Tate(q)/C))−1 everywhere.

In summary, we have the following theorem.

Theorem 23 Let R be a Zp-flat p-adic ring, and let r ∈ R be such that r|p and p/r is topologically

nilpotent in R. Let l be prime number not dividing 2Np. Then there is an endomorphism Tl2 of

the space Mk/2(4N, R, χ, r) with the property that if∑

anqn is the q-expansion of F at ∞ then the

q-expansion of Tl2F at infinity is∑

bnqn where

bn = al2n +

(−1

l

)(1−k)/2

l(k−1)/2−1(n

l

)an + lk−2χ(l2)an/l2 .

3.4 The Canonical Subgroup

In this section we assume that our ground ring R is a complete DVR with residue charac-

teristic p ≥ 5 and generic characteristic 0. We normalize the valuation v on R by v(p) = 1.

We recall the following theorem from [7], where it is attributed to Lubin.


Theorem 24 Let r ∈ R have v(r) < 1/(1+p). Then there is one and only one way to attach to each

r-test datum (E/S , Y ) a pair (H1, Y1) such that H1 is a subgroup of E of order p and (E/H1, Y1) is

an rp-test datum, such that

• the formation of (H1, Y1) is compatible with S-isomorphisms of the data (E, Y ) and with ar-

bitrary change of p-adically complete R algebra S, and

• if p/r = 0 in R, then H1 is the kernel of the Frobenius map E → E(p) and Y1 is the pull-back

of Y to E(p) = E/H1 via the dual of Frobenius.

Remark 25 The subgroup H1 in the above Theorem is called the canonical subgroup of E. Morally,

this result states that canonical subgroups exist for elliptic curves which are “not too supersingular”

and taking the quotient of an elliptic curve by its canonical subgroup makes it more supersingular

(unless it is actually ordinary). If E = Tate(q) over the base R/pnR, then the section Y making

(E, Y ) an r-test datum is unique and Katz notes that the above correspondence assigns the subgroup

H1 = µp to this pair.

This construction can be iterated to obtain canonical cyclic subgroups of order pn for

n = 1, 2, . . . . That is, one can construct a group H2 as the inverse image of the canonical subgroup

of E/H1 in the natural map E → E/H1, etc. The result of this analysis is summarized in the

following theorem, which is essentially Theorem II.2.7 of [5] (the only difference being that Gouvea

doesn’t keep track of the sections Yn at each stage, which is why his restriction on r is weaker).

Theorem 26 Let r ∈ R have v(r) < 1/pn−1(1 + p). To each r-test datum (E/S , Y ) we may attach

a pair (Hn, Yn) consisting of a cyclic subgroup of order pn and a section of ω1−p(E/Hn)/S such that

(E/Hn, Yn) is an rpn

-test datum, in such a manner that

• (Hn, Yn) depends only on the isomorphism class of (E/S , Y ) and is compatible with arbitrary

change of p-adically complete R-algebra S,

• if p/rpn−1

= 0 in S, then Hn is the kernel of the iterated Frobenius

E → E(p) → · · · → E(pn),

• if E = Tate(q) (and Y is the unique section making (Tate(q), Y ) an r-test datum), then

Hn = µpn .

Proof. The case n = 1 is Theorem 24 above (and the comment which follows it). Suppose that

the result is true for n − 1. Thus we may attach to (E, Y ) a pair (Hn−1, Yn−1) as indicated with

(E/Hn−1, Yn−1) an rpn−1

-test datum. Since v(rpn−1

) = pn−1v(r) < 11+p we may apply Theorem

24 to this pair to get a subgroup Hn of E/Hn−1 and a section Yn making ((E/Hn−1)/Hn, Yn) an

rpn

-test datum. We simply let Hn be the inverse image of Hn in E. All the properties follow readily

from Theorem 24. See [5] for further details.


3.5 The operator Up2

In this section we continue to assume that R is a complete DVR of mixed characteristic

and residue characteristic p ≥ 5. Let K denote the field of fractions of R. Our aim in the section is

to prove the following Theorem.

Theorem 27 Let v(r) < 1/p(1 + p). Then there exists a continuous endomorphism Up2 of the

Banach space Mk/2(4N, R, r)⊗K with the following properties.

• ‖Up2‖ ≤ p2,

• Up2 acts as∑

anqn 7−→∑

ap2nqn

on q-expansions at ∞, and

• if v(r) > 0, then Up2 is completely continuous.

Suppose that r ∈ R has v(r) < 1/p(1+p). Then the results of the previous section, together

with a gluing argument similar to that in Section 3.3, give a compatible family of degeneracy maps

πn : X1(4N)(r)n −→ X1(4N)(rp2

)n

gotten by taking the quotient by the canonical subgroup of order p2.

In constructing the operator Up2 we would like to proceed as in the construction of other

Hecke operators by using the trace πn∗ of these maps on rational functions. However, one encounters

the technical hitch that there is no reason to expect the maps πn to be flat, and one can only expect

to take the trace through finite flat maps. Intuitively, one would like to pass to the generic fiber to

ensure flatness (but eschew integrality) and proceed from there. This makes no sense in the current

context as the spaces above are schemes over R/pnR and have no generic fiber. To recover a sort

of generic fiber, one would like a moduli space over R for our problem, which does not exist in the

category of schemes.

To remedy this, one proceeds as in the integral weight case by passing to rigid analytic

spaces. We denote by X1(4N) the rigid analytic space over K associated to the algebraic curve

X1(4N)/K . For each r ∈ R, one can consider the locus X(4N)(r) in this curve defined by the

condition |Ep−1| > |r|. Moreover, this locus is an affinoid subdomain of X1(4N). See Sections 1 and

2 of [2] for the details behind these statements.

According to Katz (proof of Theorem 3.10.1 in [7]), the space of analytic functions on

X1(4N)(r) coincides with the space of r-overconvergent modular forms of weight 0 and level 4N ,

tensored with K, that is,

M0(4N, R, r)⊗K = H0(X1(4N)(r), O)


in the obvious notation. More generally, Coleman ([2], [3]) generalizes this to higher weight in the

obvious way using the analytic sheaf ω, so that

Mk(4N, R, r)⊗K = H0(X1(4N)(r), ωk).

The divisor Σ4N,k restricts to X1(4N)(r), and will be denoted by the same name. In view of

our definition of r-overconvergent modular form of half-integral weight and the identification above

(in weight 0), we see that there is an identification

Mk/2(4N, R, r) = H0(X1(4N)(r), Σ4N,k),

where the H0 is interpreted as meromorphic functions on X1(4N)(r) with divisor ≥ −Σ4N,k.

For v(r) < 1/(1 + p), Theorem 24 defines a map (Frobenius)

φ : M0(4N, R, rp) −→M0(4N, R, r).

The following is Theorem 3.10.1, part III, of [7].

Theorem 28 For any r with v(r) < 1/(1 + p), the map

φ⊗K : M0(4N, R, rp)⊗K −→M0(4N, R, r)⊗K

is finite and etale of rank p.

This map of affinoid algebras corresponds to a degeneracy map

d : X1(4N)(r) −→ X1(4N)(rp)

between the corresponding affinoids (which we think of as taking the quotient by the canonical

subgroup). The upshot of the above theorem is that we now have a trace map d∗ on analytic and

meromorphic functions. Moreover, if F is any coherent analytic sheaf on X1(4N)(rp), then we also

have a trace map

d∗ : d∗d∗F −→ F.

By applying this to the sheaf O(D) for some divisor D, we see that if F is a meromorphic function

on X1(4N)(r) with (F ) ≥ −d∗D, then (d∗F ) ≥ −D. This also holds for divisors with rational

coefficients, as one can see by replacing such a D by ⌊D⌋ (as in the argument in the beginning of

Section 2.7).

Now suppose that v(r) < 1/p(1 + p). Then the composite

X1(4N)(r) −→ X1(4N)(rp) −→ X1(4N)(rp2

)

is finite and etale of rank p2, and will be denoted by π. Theorem 24 with n = 2 allows us to

regard X1(4N)(r) as an affinoid subdomain of the rigid space X(4N ; p2) associated with the curve


X1(4N ; p2)/K (this is done in Section 6 of [2] in the case n = 1). Moreover, by definition of our

degeneracy map π1 (Section 2.7), we have a commutative diagram

X1(4N)(r)i

//

π

X(4N ; p2)

π1

X1(4N)(rp2

)j

// X1(4N)

where i and j are the natural inclusions.

Consider the (meromorphic) function Θp2 on X(4N ; p2). By Section 2.6 we have

(Θkp2) = π∗

1Σ4N,k − π∗2Σ4N,k.

We will denote the restrictions of Θp2 and Σ4N,k to X1(4N)(r) by the same names. Then, as a

function on X1(4N)(r), Θkp2 has divisor

(Θkp2) = i∗(π∗

1Σ4N,k − π∗2Σ4N,k) = π∗Σ4N,k − Σ4N,k

since π1 i = j π and π2 i is the natural inclusion of X1(4N)(r) in X1(4N).

For d ∈ (Z/(4N)Z)×, let 〈d〉 denote the automorphism of X1(4N) sending (E, P ) to

(E, dP ). Note that this group of automorphisms acts trivially on X1(4) and hence preserves the

divisor Σ4N,k. Let F ∈Mk/2(4N, R, r). The corresponding meromorphic function on X1(4N)(r) has

(F ) ≥ −Σ4N,k, so that (〈d〉∗F ) ≥ −Σ4N,k as well, for any d prime to 4N . By the calculation in the

previous paragraph, the function 〈p2〉∗F ·Θ−kp2 has divisor at least

−Σ4N,k − (π∗Σ4N,k − Σ4N,k) = −π∗Σ4N,k.

It follows from the above discussion of traces that π∗(〈p2〉∗F · Θ−k

p2 ) has divisor at least −Σ4N,k.

Thus we have a well-defined map

Mk/2(4N, R, r)⊗K −→ Mk/2(4N, R, rp2

)⊗K

F 7−→ p−2π∗(〈p2〉∗F ·Θ−k

p2 )

We define Up2 to be the composition of this map with the natural map

Mk/2(4N, R, rp2

)⊗K −→Mk/2(4N, R, r)⊗K.

The pair (Tate(q), ζ4N ) defines a Z((q4N ))⊗K[ζ4N ]-valued point on X1(4N)/K , and since

Tate(q) is ordinary, it gives such a point on X1(4N)(rp2

). Let d be an inverse of p2 mod 4N . The

fiber of π1 above this point consists of all triples (Tate(q)/C, dP/C, ker(pC)), where C runs through

the cyclic subgroups of order p2, and

pC : Tate(q)→ Tate(q)/C


is the natural map. The fiber of π above this point is the subcollection consisting of those triples in

which ker(pC) is the canonical subgroup of Tate(q)/C. All of the triples

(Tate(q)/〈ζip2qp2〉, ζd

4N/〈ζip2qp2〉, µp2), i = 0, 1, . . . , p2 − 1

have this property, and there are p2 of them, so this must be all of them.

Suppose F ∈Mk/2(4N, R, r) has q-expansion∑

anqn at ∞. Then

(Up2F )(Tate(q), ζ4N )

= p−2π∗(〈p2〉∗F ·Θ−k

p2 )(Tate(q), ζ4N )

= p−2

p2−1∑

i=0

(〈p2〉∗F )(Tate(ζip2qp2), ζd

4N )Θp2(Tate(ζip2qp2), ζd

4 )−k

= p−2

p2−1∑

i=0

F (Tate(ζip2qp2), ζ4N )

( ∑n∈Z

qn2

∑n∈Z

(ζip2qp2)n2

)−k

= p−2 1

(∑

n∈Zqn2)k

p2−1∑

i=0

∑an(ζi

p2qp2)n =

∑ap2nqn

(∑

n∈Zqn2)k

,

and hence the q-expansion of Up2F at ∞ is∑

ap2nqn.

We now take up the issue of boundedness of Up2 . The Banach space F0(4N, R, r) ⊗ K

coincides with the meromorphic functions on X1(4N)(r) which are analytic outside of the cusps,

and is the affinoid algebra corresponding to the complement of the cuspidal locus in X1(4N)(r). As

the restriction of the map π to this locus is still finite and etale of rank p2, there is a trace map d∗

in this context as well.

Proposition 29 The trace map

d∗ : F0(4N, R, r)⊗K −→ F0(4N, R, rp)⊗K

has ‖d∗‖ ≤ 1.

Proof. It suffices to verify this after extending scalars to some larger field. Let K denote the

completion of an algebraic closure of K and let R∞ denote the ring of integers in K. An explicit

calculation carried out in the proof of Proposition II.3.6 of [5] shows that, over R∞, one can actually

take the trace of a rule in F0(4N, R, r) to obtain a new rule in F0(4N, R, rp). This is because over

the ring R∞ one can solve explicitly for the triples (E′, P ′, Y ′) mapping to a particular (E, P, Y )

via “quotient by the canonical subgroup” as in Theorem 24. In particular,

d∗(F0(4N, R∞, r)) ⊂ F0(4N, R∞, rp),

which implies the result.


For any r ∈ R, the function Θp2 has norm 1 in F0(4N, R, r)⊗K, as both Θp2 and Θ−1p2 lie

in the unit ball F0(4N, R, r). Thus the composite

Mk/2(4N, R, r)⊗K·Θ−k

p2

−→ F0(4N, R, r)⊗Kπ∗−→ F0(4N, R, rp2

)⊗K

has norm at most one, and its image lies in Mk/2(4N, R, rp2

). Further composing with the natural

map

Mk/2(4N, R, rp2

)⊗K −→Mk/2(4N, R, r)⊗K

gives the operator p2Up2 . As this last map is completely continuous when v(r) > 0 (because then

the inclusion X1(4N)(rp2

) ⊂ X1(4N)(r) is inner in the sense of [3], Section A.5), we finally arrive

at Theorem 27.

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