SUPERSINGULAR ELLIPTIC CURVES, THETASERIES AND WEIGHT TWO MODULAR FORMS

Matthew Emerton
Northwestern University

This paper deals with two subjects and their interaction. The first is the problemof spanning spaces of modular forms by theta series. The second is the commutative
algebraic properties of Hecke modules arising in the arithmetic theory of modular
forms.
Let p be a prime, and let B denote the quaternion algebra over Q that is ramified
at p and ∞ and at no other places. If L is a left ideal in a maximal order of B
then L is a rank four Z-module equipped in a natural way with a positive definite
quadratic form [6, §1]. (We shall say that L is a rank four quadratic space, andremark that the isomorphism class of L as a quadratic space depends only on theleft ideal class of L in its maximal order.) Eichler [5] proved that the theta seriesof L is a weight two modular form on Γ0 (p), and that as L ranges over a collectionof left ideal class representatives of all left ideals in all maximal orders of B these
theta series span the vector space of weight two modular forms on Γ0 (p) over Q.
In this paper we strengthen this result as follows: if L is as above, then the
q-expansion of its theta series Θ(L) has constant term equal to one and all othercoefficients equal to even integers. Suppose that f is a modular form whose qexpansion coefficients are even integers, except perhaps for its constant term, which
we require merely to be an integer. It follows from Eichler’s theorem that f may
be written as a linear combination of Θ(L) (with L ranging over a collection ofleft ideals of maximal orders of B) with rational coefficients. We show that in fact
these coefficients can be taken to be integers.
Let T denote the Z-algebra of Hecke operators acting on the space of weight two
modular forms on Γ0 (p). The proof that we give of our result hinges on analyzing
the structure of a certain T-module X . We can say what X is: it is the free Zmodule of divisors supported on the set of singular points of the (reducible, nodal)
curve X0 (p) in characteristic p. The key properties of X , which imply the above
result on theta series, are that the natural map T −→ EndT (X ) is an isomorphism,
and that furthermore X is locally free of rank one in a Zariski neighbourhood of
the Eisenstein ideal of T. We remark that it is comparatively easy to prove theanalogous statements after tensoring with Q, for they then follow from the factthat X is a faithful T-module. Indeed, combining this with the semi-simplicity of
the Q-algebra T⊗Z Q, one deduces that X ⊗Z Q is a free T⊗Z Q-module of rank one,and in particular that the map T ⊗Z Q −→ EndT⊗Z Q (X ⊗Z Q) is an isomorphism.
2000 Mathematics Subject Classification: 11F11, 11F27, 11F37
Typeset by AMS-TEX
1

2

MATTHEW EMERTON

By contrast, X need not always be locally free of rank one as a T-module. Indeed,
we show that this is the case if and only if T is a Gorenstein ring, and Kilford [14]has shown that T can be non-Gorenstein (for example, when p equals 431 or 503).
The connection between Eichler’s theorem and the T-module X was observedby Ohta [18] (actually, he considered only the mod p reduction of X ), and further
developed by Gross [6]. It was Gross who raised the question of studying thestructure of X and of strengthening Eichler’s result [8]. In addition to the ideas of[6], our arguments will rely heavily on the results and techniques of [15] and [20, 23],
which provide powerful tools for analyzing questions related to the arithmetic ofmodular forms.
The contents of the paper are as follows. Section 0 presents a more detailed
introduction to the paper, and states the main results. It concludes with a complete
analysis of the case p = 11, to serve the reader as an example. Sections 1, 2, 3 and4 develop all the inputs necessary for the proofs of the results stated in section 0.
The actual proofs of these results are presented in section 5. Section 6 discusses
some aspects of the situation when T is non-Gorenstein; in particular, it provides aninterpretation of some of our results in the language of “local theta-characteristics”
of one-dimensional local rings. Section 7 discusses the extension of our results to thecase of non-prime level. Finally, the appendix relates the analysis of the T-module
X made in the main body of the paper to the p-adic analytic properties of X0 (p);it is somewhat disjoint from the rest of the paper.
We close this introduction by remarking on two novel points that arise in the
proof of our main results. The first is that we obtain in sections 3 and 7 a new proof
that the theta series of various quadratic forms constructed from the quaternion
algebra B are modular forms, which is algebro-geometric rather than analytic innature. The second is theorem 4.6, which yields some information about the torsion
subgroup of a modular Jacobian in the absence of a Gorenstein hypothesis, byappealing instead to level lowering results. We hope that both ideas may haveadditional applications.
Acknowledgments. The author would like to thank Benedict Gross for explaining the problem discussed in this paper to him in the Spring of 1996, and Shuzo
Takahashi for reminding him of the problem in the Fall of 1999. He would also liketo thank Robert Coleman for interesting conversations on related topics during theSummer of 1999, Fred Diamond for a suggestion which simplified the proof of proposition 1.8 (i), Lloyd Kilford for communicating the examples of non-Gorenstein local
Hecke rings, Sándor Kovács for providing the example involving normalization that
follows definition 6.1, Ken Ribet for encouraging him to treat fully the situation at
two and three and to state a result for cusp forms, and Mike Roth for numerous enlightening conversations and computations. Finally, thanks go to Kevin Buzzard,
Benedict Gross, Barry Mazur, Ken Ribet, Mike Roth, Shuzo Takahashi and theanonymous referee for their helpful comments on various preliminary versions of
this paper, which helped to improve both the mathematics and the exposition.
0. Statement of results.
We will describe our results in more detail. We begin by introducing somenotation and filling in some background. If f is a weight two modular form on Γ0 (p),∞X
we let an (f ) denote the nth q-expansion coefficient of f , so that f =an (f )q n .
n=0

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS

3

We let M denote the Z-module of weight two modular forms f on Γ0 (p) for which1an (f ) ∈ Z when n ≥ 1 and a0 (f ) ∈ Z.
2
One way of producing elements in M is as follows: if Ei and Ej are two supersingular elliptic curves in characteristic p, then Li,j = Hom(Ei , Ej ) is a free Z-moduleof rank four equipped with a positive definite quadratic form, given by taking thedegree of an isogeny (see proposition 3.6 below for details). If Θ(Li,j ) denotes the1theta series of Li,j , then Θ(Li,j ) lies in M. (See proposition 3.15 below, as well as
2
the subsequent remarks, for a discussion of this point.) The quadratic spaces Li,jare precisely those attached to the quaternion algebra B in the manner describedin the introduction (as is explained in [6, §2]).
Let X0 (p) denote the curve over Spec Z that is a coarse moduli space for theΓ0 (p)-moduli problem. The fibres of X0 (p) are smooth curves over every point ofSpec Z other than p. By contrast, if g denotes the (constant) arithmetic genus of
the fibres of X0 (p) (which we recall is approximately equal to p/12, as follows fromproposition 3.15 below, for example), then the geometric fibre of X0 (p) in characteristic p is the union of two rational curves meeting at g +1 ordinary double points.
Let x0 , . . . , xg denote these singular points; they are in bijective correspondencewith the isomorphism classes of supersingular curves Ei . Let X denote the freeZ-module of divisors supported on the xi , and define a Z-bilinear pairing(0.1)

X × X −→ M

1by the formula xi × xj 7→ Θ(Li,j ).
2
As above, we let T denote the Z-algebra of Hecke operators acting on the spaceof weight two modular forms on Γ0 (p). The Z-module M is closed under theaction of T, and so M is naturally a faithful T-module. The action of the Heckecorrespondences on the set of xi induces an action on X which factors through T,
making X a faithful T-module as well (see theorem 3.1 below). One shows (see [6]or section 3 below) that the pairing (0.1) is T-bilinear, and so induces a map(0.2)

X ⊗T X −→ Mfrom the tensor product of X with itself over T to M.
Theorem 0.3. The morphism (0.2) is surjective. Equivalently, M is spanned by1
the half-theta series Θ(Li,j ).
2
Recall that X and M are both free of rank g + 1 over Z (where g is equal
to the genus of X0 (p)), and so in particular (taking into account the symmetry∼
Li,j −→ Lj,i ; see lemma 3.7) there are (g + 1)(g + 2)/2 theta series Θ(Li,j ). Thus
although theorem 0.3 yields an explicit spanning set for M as a Z-module, it does
not yield a basis for M (except in the case when g = 0).
As we indicated in the introduction, theorem 0.3 is closely connected to thefollowing result.
Theorem 0.4. The natural morphism T −→ EndT (X ) is an isomorphism.
The passage between the intrinsic structure of the T-module X and the surjectivity of (0.2) is provided by the duality between Hecke operators and modular forms

4

MATTHEW EMERTON(discussed in section 1), some commutative algebra of bilinear pairings (discussed
in section 2), and the existence of a natural Z-valued T-bilinear pairing on X . If
we let ei denote one-half the number of automorphisms of Ei , then this pairing
is defined as follows: hxi , xj i = ei δi,j . The connection with (0.2) is provided by1
the formula hxi , xj i = a1 ( Θ(Li,j )). (See section 3 for a more complete discussion
2of this pairing, and proposition 3.11 and the surrounding text in particular for a
precise statement of its connection with (0.2).)
As already mentioned in the introduction, the T-module X is not always locally
free of rank one. In order to state our results concerning this subject, we introduce
some additional notation.
We let M0 denote the T-submodule of M consisting of those f for which a0 (f ) =0 (that is, the submodule of cusp forms in M), and let T0 denote that quotientof T that acts faithfully on M0 . We let X 0 denote the submodule of X consistingof divisors of degree zero. Then X 0 is also a faithful T0 -module (see theorem 3.1
below).
For any maximal ideal m of T and any T-module U we let Um denote the completion of U at m. In particular, Tm denotes the completion of T at m, and Um isnaturally a Tm -module.
Theorem 0.5. Fix a maximal ideal m of T. The following are equivalent:(i) The Tm -module Mm is free of rank one.
(i)0 The T0m -module M0m is free of rank one.
(ii) The Tm -module Xm is free of rank one.
(ii)0 The T0m -module Xm0 is free of rank one.
(iii) The ring Tm is Gorenstein.
(iii)0 The ring T0m is Gorenstein.
If these equivalent conditions hold, then the morphism Xm ⊗Tm Xm −→ Mm
induced by (0.2) is an isomorphism.
It seems worth remarking on the nature of the Gorenstein hypotheses that appear
among the equivalent statements of theorem 0.5. It follows from the results of [15]that Tm and T0m are Gorenstein in most cases (see theorem 1.14 below for the precise
statement). In particular this is the case if m is Eisenstein (in the sense describedin section 1 below). The proof of this result depends on the deepest results of[15], and we rely upon these same results to prove theorem 0.5 in the Eisensteincase. In particular, in this case theorem 0.5 should properly be regarded, both inits statement and its proof, as showing that “if m is an Eisenstein maximal ideal,
then the following properties hold”, rather than as presenting a list of equivalent
properties that may or may not hold.
In the non-Eisenstein case the situation is quite different. Interestingly, and perhaps contrary to general expectations, L. Kilford [14] has recently found examplesof non-Eisenstein m for which Tm is non-Gorenstein. (The results of [15] show thatsuch m are necessarily of residue characteristic two and ordinary.) Thus in this casetheorem 0.5 is genuinely a list of equivalent possibilities, that may or may not holdin any particular case. (It is mentioned in [14] that William Stein has implemented
theorem 0.5 as a means of testing whether or not Tm is Gorenstein, by taking advantage of the fact that condition (ii) is amenable to being checked by computer
algebra.)SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS

5

Since the modules X and X 0 appear with equal status among the conditions oftheorem 0.5, one might ask whether suitable analogues of theorems 0.3 and 0.4 holdwhen X is replaced by X 0 and M by M0 . We will state a result that answers thisquestion, after making a definition. Let TEis denote the quotient of T which actsfaithfully on the weight two Eisenstein series on Γ0 (p) (so that after forgetting theT-algebra structure, TEis is isomorphic to Z). Let I denote the ideal in T which is
the kernel of the surjection T −→ TEis . Then the ideal IT0 of T0 has finite index
in T0 , equal to the numerator of (p − 1)/12 (when written in lowest terms) (seeproposition 1.8, part (iii)); denote this number by n. Note that this ideal IT0 is
the celebrated Eisenstein ideal of [15].
Theorem 0.6. (i) The image of the morphism X 0 ⊗T0 X 0 −→ M0 induced by(0.2) is equal to IM0 , and is of index n in M0 .
(ii) The natural map T0 −→ EndT0 (X 0 ) is an isomorphism.
As a final variant on our theme, we note that the Z-valued pairing on X describedabove yields an embedding X −→ HomZ (X , Z), whose image has finite index in
HomZ (X , Z). Thus we may extend the pairing (0.1) to a pairing(0.7)

X × HomZ (X , Z) −→ M ⊗Z Q.

If x̌0 , . . . , x̌g denotes the basis of HomZ (X , Z) dual to the basis x0 , . . . , xg of X ,xjthen x̌j =for j = 0, . . . , g, and so the pairing (0.7) maps the pair (xi , x̌j ) toej
Θ(Li,j ). Thus, if we let N denote the submodule ofthe normalized theta series
2ej
M ⊗Z Q consisting of modular forms f for which an (f ) ∈ Z if n ≥ 1 (but with nocondition on a0 (f )), then (0.7) takes values in N , and induces a morphism(0.8)

X ⊗T HomZ (X , Z) −→ N .

There is an analogue for X 0 , yielding a morphism(0.9)

X 0 ⊗T0 HomZ (X 0 , Z) −→ M0 .

(Note that there is no distinction between cusp forms in M and in N .) The
following result describes the images of these morphisms.
Theorem 0.10. The morphisms (0.8) and (0.9) are surjections of T-modules.
Note that the surjectivity of (0.8) is equivalent to the statement that N is
Θ(Li,j )spanned over Z by the normalized theta series.
2ej
We owe the idea of studying the T-module X and the map (0.2) entirely toGross, and the preceding results answer (in a slightly refined form) questions raisedby him in [8]. In these lectures Gross also pointed out that a positive answer
to these questions would yield a formal analogy with a theorem of Hecke in algebraic number theory [11, thm. 177], which we now explain. Namely, it follows
from theorem 1.14 below that T becomes Gorenstein after inverting 2, and frompropositions 1.1 and 1.3 that there is a natural injection M −→ HomZ (T, Z) whose
cokernel has order equal to the denominator of (p − 1)/12, which hence becomes an

6

MATTHEW EMERTON

isomorphism after inverting 6. Theorem 0.5 thus shows that after inverting 6 themorphism (0.2) yields an isomorphism of locally free T[1/6]-modules of rank one∼∼

X [1/6] ⊗T[1/6] X [1/6] −→ M[1/6] −→ HomZ[1/6] (T[1/6], Z[1/6]).
In particular, HomZ[1/6] (T[1/6], Z[1/6]) is a square in the Picard group of T[1/6].
By way of comparison, Hecke’s theorem states that if O is the ring of integers ina finite extension of Q, then the inverse different D−1 of O (which by definition
is isomorphic to HomZ (O, Z) as an O-module) is a square in the ideal class group(that is, the Picard group) of O.
One can ask if the analogue of Hecke’s result holds without inverting 6. That is,in those cases when T is Gorenstein, whether the T-module HomZ (T, Z) is a square
element in the the Picard group of T; or whether HomZ[1/2] (T[1/2], Z[1/2]) is asquare element in the Picard group of T[1/2] (remembering that T[1/2] is always
Gorenstein). We don’t know the answer to either of these questions in general.
An example. We will describe our results in as concrete a fashion as possible
in the case when p = 11. In particular, we will describe T explicitly, identify theelements of Pic(T) corresponding to M, N , X , and HomZ (X , Z), and describe themaps (0.2) and (0.8) in terms of these elements of Pic(T).
We begin by recalling that the space of modular forms of weight two on Γ0 (p)5+ q + 3q 2 + · · · andis two-dimensional, spanned by the Eisenstein series E =12Q∞
the cusp form f = q n=1 (1 − q n )2 (1 − q 11n )2 = q − 2q 2 + · · · . These two Hecke
eigenforms are congruent modulo five, and so one sees that M is the free Z-moduleE−f6(E − f )spanned by
and f , while N is the free Z-module spanned by
and
5
5
f.
We now provide an explicit description of the Hecke ring T. Each of TEis and0
T is isomorphic to Z, and (since E and f are congruent modulo five) the image ofT under the injection T −→ TEis ⊕ T0 is equal to {(m1 , m2 )|m1 ≡ m2 (mod 5)}.
This ring is Gorenstein; indeed, the pairing(m1 , m2 ) × (m01 , m02 ) 7→

m1 m01 − m2 m02
5yields an isomorphism of T with its Z-dual. (Since f is supersingular at two, this isa special case of theorem 1.14 below.) One computes that Pic(T) is isomorphic tothe cokernel of the natural map Z× −→ (Z/5)× , which has order two. The module
U = {(m1 , m2 )|2m1 ≡ m2 (mod 5)} (with the obvious T-action) represents the
non-trivial element of Pic(T).
6E
fDefine a map of T-modules T −→ M via (m1 , m2 ) 7→ m1− m2 . This5
5
is evidently an isomorphism of T-modules (given the above description of M),
and shows that M represents the trivial element of Pic(T). Similarly the map
Ef(m1 , m2 ) 7→ m1 − m2 is an isomorphism of T with N , showing that N also
55represents the trivial element of Pic(T). (This last statement also follows frompart (i) of proposition 1.3 below, together with the preceding observation that T isisomorphic to its Z-dual.) With respect to these two isomorphisms, the inclusion
M ⊂ N corresponds to the endomorphism of T defined via (m1 , m2 ) 7→ (6m1 , m2 ).

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS

7

The two supersingular j-invariants in characteristic 11 are j = 0 and j = 1728.
We denote the corresponding basis elements of X by x0 and x1 . Then e0 = 3 ande1 = 2. Write x = 2x0 + 3x1 . It follows from the results of section 3 below that
T acts on x through TEis and on x1 − x0 through T0 . Define a map of T-modulesxx1 − x0
U −→ X via (m1 , m2 ) 7→ m1 + m2
. It is immediate that this map is an
55
isomorphism, and so X represents the non-trivial element of Pic(T).
Let x̌0 and x̌1 denote the basis of HomZ (X , Z) which is dual to the basis x0 , x1of X . Then the degree morphism deg : X −→ Z is equal to the element x̌0 + x̌1 of
HomZ (X , Z), while the element 2x̌1 −3x̌0 annihilates x. It follows from the results ofsection 3 below that T acts on deg through TEis and on 2x̌1 −3x̌0 through T0 . Thusdeg
2x̌1 − 3x̌0
the morphism U −→ HomZ (X , Z) defined via (m1 , m2 ) 7→ m1+ m2
5
5
is an isomorphism of T-modules, and HomZ (X , Z) also represents the non-trivial
element of Pic(T).
∼∼With respect to the isomorphisms U −→ X and U −→ HomZ (X , Z) constructed in the preceding two paragraphs, the tautological perfect pairing X ×HomZ (X , Z) −→ Z corresponds to the perfect pairing U × U −→ Z defined by(m1 , m2 ) × (m01 , m02 ) 7→

m1 m01 + m2 m02.
5

The embedding X −→ HomZ (X , Z) induced by the Z-valued T-bilinear pairing
on X of definition 3.4 below corresponds to the endomorphism of U defined via(m1 , m2 ) 7→ (6m1 , m2 ).
Recall that Li,j denotes the quadratic space of endomorphisms from the elliptic
curve Ei representing xi to the elliptic curve Ej representing xj . From the valuesof e0 and e1 we can determine that1
6E + 9f1
Θ(L0,0 ) = + 3q + · · · =
,
2
2
5
that

1
1
6(E − f )Θ(L0,1 ) = + 0q + · · · =,225

and that1
1
6E + 4f
Θ(L1,1 ) = + 2q + · · · =.
22
5
From these formulas we compute the following particular values of the map X ⊗T
X −→ M given by (0.2):x ⊗ x 7→ 30E,
x ⊗ (x1 − x0 ) 7→ 0,and(x1 − x0 ) ⊗ (x1 − x0 ) 7→ 5f.
Since both the source and target of (0.2) are torsion free Z-modules, these values
completely determine this map. One finds that, in terms of the given isomorphisms∼∼∼
U −→ X and T −→ M, the map (0.2) corresponds to the isomorphism U ⊗T U −→T, given by(0.11)(m1 , m2 ) ⊗ (m01 , m02 ) 7→ (m1 m01 , −m2 m02 ),8

MATTHEW EMERTON

which witnesses the fact that U yields an element of Pic(T) of order two. We also
see that the induced map X 0 ⊗T0 X 0 −→ M0 = Z · f has image equal to 5Z · f .
(Note that 5 is the numerator of (11 − 1)/12 = 5/6.)
Similarly, we compute the following values of the map X ⊗T HomZ (X , Z) −→ Ngiven by (0.8):
x ⊗ deg 7→ 5E,
x ⊗ (2x̌1 − 3x̌0 ) 7→ 0,(x1 − x0 ) × deg 7→ 0,and(x1 − x0 ) × (2x̌1 − 3x̌0 ) 7→ 5f.
∼

From these values we deduce that, in terms of the given isomorphisms U −→ X ,∼∼U −→ HomZ (X , Z), and T −→ N , the map (0.8) also correspond to the iso∼morphism U ⊗T U −→ T given by (0.11). If we note that HomZ (X 0 , Z) is the2x̌1 − 3x̌0
free Z-module of rank one spanned by, we also deduce that the map
5000
X ⊗T0 HomZ (X , Z) −→ M is surjective.
Taken together, the above calculations establish all our results in the case p = 11.
1. The T-module M.
As in section 0, we let M denote the Z-module of modular forms f of weight two
1
on Γ0 (p) for which an (f ) lies in Z (n ≥ 1) and a0 (f ) lies in Z, and let T denote2
the Z-algebra of Hecke operators, which (tautologically from its construction) actsfaithfully on M. Also, M0 denotes the submodule of cusp forms in M (that is,
those f for which a0 (f ) = 0), and T0 the quotient of T that acts faithfully on M0 .
The tensor product M ⊗Z Q is naturally identified with the space of weight two
modular forms on Γ0 (p) all of whose q-expansion coefficients lie in Q. We let N
denote the T-submodule of M ⊗Z Q consisting of those f for which an (f ) lies in
Z when n ≥ 1, and for notational consistency we will let N 0 denote M0 whenregarded as the submodule of cusp forms in N .
∞X
p−1 X 0Let E =+
σ (n)q n (where σ 0 (n) =d) denote the weight two
24
d|n
n=1
(p,d)=1

Eisenstein series on Γ0 (p). Then E is a T-eigenform, and so defines a surjection
T −→ Z that sends the Hecke operator Tn to the integer σ 0 (n). We let TEis denote
Z regarded as a quotient of T in this way. We denote by N Eis the T-submoduleof N spanned by the Eisenstein series E, and by MEis the intersection M ∩ N Eis .
These are both free modules of rank one over TEis , and MEis (respectively N Eis )is the maximal submodule of M (respectively N ) on which the T-action factors
through TEis .
We define the positive integers n and ∆ to be respectively the numerator anddenominator of the fraction (p−1)/12, when reduced to lowest terms. Since a0 (E) =(p − 1)/24, we see immediately that ∆E lies in M, and generates MEis .
By definition N contains M, and since both are finitely generated Z-modules
spanning the same Q-vector space, M has finite index in N . The following resultshows that this index is exactly equal to ∆.

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS

9

Proposition 1.1. Let A0 : N −→ Q denote the map f 7→ 2∆a0 (f ). Then A0
is in fact a surjection from N onto Z, which fits into the following commutative
diagram, whose rows and columns are short exact:
(1.2)

0

0

0

0


/ M0


N0


/0

/0

0


/M


/N


/ M/N

/0

0


/ ∆Z


/Z


/ Z/∆

/0


0


0


0

.

A0

∼

In this diagram, the action of T on the objects of the bottom row factors throughTEis .
Proof. If p = 2 or 3, then N = N Eis is spanned by E, and M = MEis is spannedby ∆E. Given this, the truth of the proposition in these cases is immediate. Thus
we assume for the remainder of the proof that p ≥ 5.
We first show that A0 maps N into Z. We will use the results of [15, §’s II.4, II.5],
which provide a detailed study of the arithmetic of q-expansions of weight two
modular forms on Γ0 (p). Note that in this reference B(Z) denotes the submoduleof N consisting of modular forms all of whose q-expansion coefficients lie in Z, whileB 0 (Z) denotes the module we have called N 0 . For any ring R, B(R) = B(Z) ⊗Z R
and B 0 (R) = B 0 (Z) ⊗Z R.
Suppose that f ∈ N ; write a0 (f ) = a/b, where a and b are coprime integers.
Then b(p −1)f −24aE is an element of N 0 , which is congruent to −24aδ (mod (p −1)b). (Here δ is the formal q-expansion introduced in [15, p. 78].) Thus −24aδ liesin B 0 (Z/(p − 1)b), and from [15, prop. II.5.12] we deduce that (p − 1)b divides 24an.
Writing (p − 1) = n(12/∆), we see that this is equivalent to b dividing 2∆a; sincea and b are coprime, we deduce that b divides 2∆. Thus A0 (f ) = 2∆a/b ∈ Z.
The preceding paragraph shows that A0 is a map N −→ Z. Immediately fromits definition, we see that N 0 is the kernel of A0 . In order to complete the proof
that (1.2) has exact rows and columns we have to show that A0 yields a surjectionof N onto Z, and restricts to yield a surjection of M onto ∆Z.
Note that the Eisenstein series E maps to the integer
A0 (E) = 2∆a0 (E) = 2∆

p−1
= n.
24

On the other hand we can find g ∈ M such that a0 (g) = 1/2 (via the theta series
construction of section 3, for example), and so
A0 (g) = 2∆a0 (g) = ∆.

10

MATTHEW EMERTON

Thus we see that the image of M does equal ∆Z, while the image of N contains
g.c.d(n, ∆) = 1, and so equals Z.
Finally, we have to show that the action of T on the bottom row factors throughTEis . This is perhaps most rapidly seen by tensoring the diagram through with Q∼
over Z, and using the isomorphism N Eis ⊗Z Q ⊕ N 0 ⊗Z Q −→ N ⊗Z Q. 
One point of view on the preceding result is as follows: [15, cor. II.5.11 (ii)] showsthat if p ≥ 5 and m is an integer prime to p, then 1 is the q-expansion of a modular
form for Γ0 (p) over Z/m if and only if m divides 12. However, this modular form
may not lift to a modular form in characteristic zero. (In the terminology of [15],it is a modular form in ω 2 , but not necessarily in B(Z/m).) Proposition 1.1 showsthat 1 is the q-expansion of an element of B(Z/m) if and only if m divides 2∆.
We will require the duality between modular forms and Hecke operators, which
we recall in the following proposition.
Proposition 1.3. (i) There is a natural isomorphism of T-modules∼

N −→ HomZ (T, Z),
defined by the pairing hf, T i = a1 (f |T ).
(ii) There is a natural isomorphism of T0 -modules∼

M0 = N 0 −→ HomZ (T0 , Z),
defined by the same pairing as that of part (i).
Proof. The assertion of (ii) is obtained by specializing [19, thm. 2.2] to the case ofΓ0 (p). A simple modification of the proof of this theorem yields (i). The key pointis that a weight two modular form cannot have a constant q-expansion. 
We now digress slightly, and present a point of view on the results of [15, p. 96]
which is in keeping with the arguments of this section.
Consider the injection(1.4)

0 −→ T −→ TEis ⊕ T0 .

This is Z-dual to the injection(1.5)

0 −→ N Eis ⊕ N 0 −→ N .

We also have the short exact sequence
(1.6)

A

0
0 −→ N 0 −→ N −→Z −→ 0which makes up the central column of (1.2).
The map A0 : N −→ Z corresponds by proposition 1.3 to an element of T, which
we denote by t0 , characterized by the property a1 (f |t0 ) = A0 (f ) = 2∆a0 (f ) for
any f ∈ N . Dualizing (1.6) yields the short exact sequence(1.7)

0 −→ Zt0 −→ T −→ T0 −→ 0.

We now prove a slightly rephrased version of [15, prop. II.9.7].

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 11

Proposition 1.8. (i) The kernel of the surjection T −→ T0 is principal, generated
by t0 .
(ii) The element t0 − n of T lies in the kernel of the surjection T −→ TEis .
(iii) The cokernel of the injection (1.4) is cyclic of order n.
Proof. Claim (i) follows from the exact sequence (1.7). Claim (ii) follows from thefact that A0 (E) = 2∆a0 (E) = n, while a1 (E) = 1. To prove claim (iii), it suffices
to prove the same statement for the cokernel of the injection (1.5). A considerationof the short exact sequence (1.6) shows that the cokernel of (1.5) is isomorphic to
A0 (N )/A0 (N Eis ) = Z/A0 (E) = Z/n. This proves the proposition. 
The idea of using the surjectivity of A0 to prove statement (i) of the previous proposition arose out of a suggestion of Diamond, and simplifies our original
argument.
From proposition 1.8 (iii) we see that the natural injection T −→ TEis ⊕ T0
is not an isomorphism unless n = 1 (in which case p = 2, 3, 5, 7 or 13, and so
T0 = 0); this is the main theme of [15]. We regard Spec T as being the union of
its closed subschemes Spec TEis and Spec T0 . From this optic, the maximal ideals
m of T are seen to be of three types. The first type consists of those that lie inSpec T \ Spec TEis ; that is, they are the maximal ideals that induce the unit idealof TEis , but a non-unit ideal of T0 . The second type consists of those that lie inSpec T \ Spec T0 ; that is, they are the maximal ideals that induce the unit idealof T0 , but a non-unit ideal of TEis . The third type consists of those that lie in
the intersection of Spec TEis and Spec T0 ; that is, they are the maximal ideals thatinduce a non-unit ideal of both TEis and T0 (the primes of fusion between TEis andT0 in T). We refer to the maximal ideals of this third type as the Eisenstein maximal
ideals of T. From proposition 1.8 (iii) we deduce that the residue characteristicsof the Eisenstein primes are precisely the prime numbers that divide n. (Thiscalculation was first carried out in [15, p. 96].) This will be quite important in thearguments to come, as we see already in the next proposition.
Proposition 1.9. For any maximal ideal m of T, the completion Mm is free ofrank one as a Tm -module if and only the completion Nm is free of rank one as a
Tm -module.
Proof. Let ` denote the residue characteristic of m. If m is of the first type (that is,
generates the unit ideal of TEis ), then completing (1.2) at m yields an isomorphism∼
Mm −→ Nm , and the corollary is clear in this case. If m is of the second type (thatis, generates the unit ideal of T0 ), then completing (1.2) at m yields isomorphisms∼∼∼∼
Mm −→ ∆Z` and Nm −→ Z` . Thus both are free Tm −→ TEism −→ Z` -modulesof rank one, so that the corollary is also clear in this case. If m is an Eisenstein
ideal, then ` divides n, and so does not divide ∆ (since n and ∆ are coprime by∼construction). Thus completing (1.2) at m yields an isomorphism Mm −→ Nm ,and the corollary follows in this case as well. The following interpretation of the results of [15] was pointed out to the author
by Gross.
Theorem 1.10. Let m be a maximal ideal of T. Then Tm is Gorenstein if andonly if T0m is Gorenstein.
∼

Proof. If m is a maximal ideal of the first type, then Tm −→ T0m , and so the theorem∼
is true in this case. If m is a maximal ideal of the second type then Tm −→ TEis
m ,

12

MATTHEW EMERTON

and so T0m = 0 (and hence is Gorenstein). On the other hand, if ` denotes the∼residue characteristic of m then TEism −→ Z` . Since this ring is also Gorenstein, thetheorem follows in this case.
It remains to treat the case when m is Eisenstein. In this case, let I denote the0 ∼kernel of the map Tm −→ TEism and J denote the kernel of the map Tm −→ Tm −→EisZ` . Then I ∩ J = 0, and so Spec Tm is the scheme-theoretic union of Spec Tm and0Spec T0m . The scheme-theoretic intersection of TEism and Tm is certainly a CartierEis ∼divisor on Spec Tm −→ Spec Z` , while [15, thm. II.18.10] implies that it is a Cartierdivisor on Spec T0m .
It is a general result that if a scheme X is the scheme-theoretic union of two
closed subschemes that meet along a common Cartier divisor, then X is Gorenstein
if and only if each of these subschemes is Gorenstein. (See lemma 1.11 below.)
Applying this in the above case, we see that Spec Tm is Gorenstein if and only if0
Eis ∼−→ Spec Z` is Gorenstein,Spec TEism and Spec Tm are Gorenstein. Since Spec T
the assertion of the theorem follows. The following result was explained to the author by Roth. We include a proof
for the sake of completeness.
Lemma 1.11. Let X be a finite-dimensional Noetherian scheme which admits adualizing complex F • . Suppose that X1 and X2 are two closed subschemes of X
such that X is the scheme-theoretic union of X1 and X2 , and such that the schemetheoretic intersection X1 ∩ X2 is a Cartier divisor on both X1 and X2 . Then X isGorenstein if and only if each of X1 and X2 are Gorenstein.
Proof. Recall that a finite-dimensional connected Noetherian scheme Y with dualizing complex FY• is Gorenstein if and only if FY• is quasi-isomorphic to a locally
free OY -sheaf of rank one supported in degree −dim(Y ) [10, §V.9]. If Y is Gorenstein and Z is a Cartier divisor on Y, then Z is also Gorenstein (this is true of
any local complete intersection in a Gorenstein scheme [10, pp. 143–144]), anddim(Z) = dim(Y ) − 1 (since any Noetherian scheme admitting a dualizing complex
is catenary [10, §V.10]). This has the consequence that if two connected Gorenstein
subschemes of a fixed ambient scheme meet along a common Cartier divisor, theymust be of the same dimension. (We will use this remark below.)
It suffices to prove the lemma after restricting our attention to a sufficiently
small neighbourhood of an arbitrary point x of X. Since it is clearly true in theneighbourhood of points in the complement of X1 ∩ X2 (since in such a neighbourhood X is simply the disjoint union of X1 and X2 ), we may in addition assume
that x lies in X1 ∩ X2 . By shrinking the neighbourhood under consideration, we
may and do assume that X1 , X2 and X1 ∩ X2 are connected.
Write D = X1 ∩ X2 . Let i1 : X1 −→ X and i2 : X2 −→ X denote the closed
immersions of X1 and X2 respectively into X. Let I1 and I2 denote the ideal
sheaves of OX cutting out X1 and X2 respectively. Then I1 ∩ I2 = 0. (This is themeaning of X being the scheme-theoretic union of X1 and X2 .) In particular, thenatural map I1 −→ (I1 +I2 )/I2 is an isomorphism. The target of this isomorphismis the ideal of the Cartier divisor D on X2 , and so is a locally free OX2 -module ofrank one. Thus I1 is a locally principal ideal sheaf in OX , with annihilator equal toI2 . Similarly, I2 is a locally principal ideal sheaf in OX , with annihilator equal to
I1 . By restricting our attention to a smaller neighbourhood of x, if necessary, we
may and do assume that both I1 and I2 are in fact principal, generated by sections

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 13

a1 and a2 of OX respectively. Thus we have a short exact sequence(1.12)·a2
0 −→ i1∗ OX1 −→OX −→ i2∗ OX2 −→ 0,

and an analogous sequence obtained by reversing the roles of X1 and X2 .
If we write F1• = i!1 F • and F2• = i!2 F • , then we obtain dualizing complexes on
X1 and X2 respectively [10, prop. V.2.4]. Applying RHom(–, F) to (1.12) (andremembering that RHom(j∗ OY , F • ) = j ! F • for any closed immersion j : Y −→ X
[10, thm. III.8.7(3)]) yields the distinguished triangle(1.13)·a2· · · −→ F2• −→ F • −→F1• −→ · · ·in the derived category of coherent OX -modules. Again, we obtain an analogous
distinguished triangle by reversing the roles of X1 and X2 .
Let us suppose that X1 and X2 are Gorenstein. The remark made at the beginning of the proof implies that X1 and X2 are of the same dimension, say n (andso X is also of dimension n). Thus F1• and F2• are quasi-isomorphic to rank one
locally free sheaves over OX1 and OX2 respectively, supported in degree −n. Taking
the long exact sequence of cohomology corresponding to the distinguished triangle(1.13), we find that F • is quasi-isomorphic to a coherent OX -module supported indegree −n, and obtain the short exact sequence·a20 −→ H−n (X2 , F2• ) −→ H−n (X, F • ) −→H−n (X1 , F1• ) −→ 0.

Reversing the roles of X1 and X2 we similarly obtain a short exact sequence·a10 −→ H−n (X1 , F1• ) −→ H−n (X, F • ) −→H−n (X2 , F2• ) −→ 0.

Combining these two exact sequence, we find that the injection H−n (X1 , F1• ) −→H−n (X, F • ) identifies H−n (X1 , F1• ) with the annihilator of a1 in H−n (X, F • ), andalso that multiplication by a2 induces a surjection of H−n (X, F • ) onto this annihilator. Again, similar remarks apply after reversing the roles of X1 and X2 .
If we write M = H−n (X, F • ), then we may summarize our conclusions by ob·a1
serving that we have a short exact sequence 0 −→ a2 M −→ M −→a1 M −→ 0,
in which a2 M is a locally free OX1 -module of rank one, and a1 M is a locally free
OX2 -module of rank one. An elementary application of Nakayama’s lemma now
implies that M is locally free of rank one over OX , and thus that X is Gorensteinin a neighbourhood of x.
The converse implication (that X Gorenstein implies X1 and X2 Gorenstein) is
proved by a similar consideration of the distinguished triangle (1.13). Since we do
not require it in this paper, we omit the details. 
Recall that Mazur has shown that the equivalent conditions of theorem 1.10 holdin almost all cases. In order to state his result precisely, we recall that if m is a
maximal ideal of T of residue characteristic `, then we say that m is ordinary if T`
does not lie in m; otherwise we say that m is supersingular. We remark that if m isof the second or third type (that is, induces a non-unit maximal ideal of TEis ) thenT` ≡ 1 + ` ≡ 1 (mod m), and so m is necessarily ordinary.

14

MATTHEW EMERTON

Theorem 1.14. The Hecke algebra T0m is Gorenstein, except possibly in the casethat m is of the first type, is ordinary and has residue characteristic two.
Proof. If m is of the first type then by [15, cor. II.15.2] we see that T0m is Gorenstein,
except in the case excluded in the statement of the theorem. If m is of the second
type, then T0m = 0 is Gorenstein. If m is Eisenstein, then [15, cor. II.16.3] showsthat T0m is Gorenstein. 
As a final result in this section, we state and prove a proposition required in the
proof of theorem 0.6.
Proposition 1.15. Let I denote the kernel of the surjection T −→ TEis . Then thequotient M0 /I is isomorphic to T0 /I; in particular, it is of order n.
Proof. We first note that T0 /I is isomorphic to the cokernel of (1.4), and so, bypart (iii) of proposition 1.8, is cyclic of order n, while the quotient M0 /I is isomorphic to M0 ⊗T0 (T0 /I). Thus both T0 /I and M0 /I are T0 -modules of finite
order annihilated by I, and hence to prove that they are isomorphic, it suffices toprove that their completions at all the maximal ideals m of T0 containing IT0 are
isomorphic. These are precisely the images in T0 of the Eisenstein maximal idealsof T, and we must thus show that M0m /I and T0m /I are isomorphic as T0m -modules
for every Eisenstein maximal ideal m of T. This follows immediately from part (ii)of proposition 1.3 together with [15, cor. II.16.3], which shows that M0m and T0m
are already isomorphic as T0m -modules. 
2. Some commutative algebra of bilinear pairings.
In this section we present some simple commutative algebra that is used in the
proof of the theorems of section 0.
We put ourselves in the following situation: A is a commutative ring with identity and B is a commutative A-algebra. We begin by recalling some adjointness
isomorphisms involving Hom and ⊗.
For the first isomorphism, let U and V be B-modules, and let W be an A-module:
then we have an isomorphism
(2.1)

∼

HomA (U ⊗B V, W ) −→ HomB (V, HomA (U, W )),defined by sending a morphism φ in the source to the morphism v 7→ (u 7→ φ(u⊗v))
in the target. For the second isomorphism, let W be a B-module: then we have an
isomorphism
(2.2)

∼

HomA (W, A) −→ HomB (W, HomA (B, A)),defined by sending a morphism φ in the source to the morphism w 7→ (b 7→ φ(bw))
in the target. (Given the preceding descriptions of these maps, it is immediate tocheck that both are indeed isomorphisms.)
Let us now fix a pair U ,V of B-modules. We have the following diagram:(2.3)

HomA (U ⊗B V, A)∼
HomB (U ⊗B V, HomA (B, A))∼/ HomB (V, HomA (U, A))∼∼ /HomB (V, HomB (U, HomA (B, A))).

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 15

The left (respectively right) vertical arrow of this diagram is obtained from (2.2)
by taking W to be U ⊗B V (respectively U ). The top arrow is obtained from (2.1)
by taking W to be A, and the bottom arrow is obtained from (2.1) by replacing A
with B and taking W to be HomA (B, A). One easily checks that (2.3) commutes.
By the definition of tensor product, the elements of HomA (U ⊗B V, A) correspond to B-bilinear pairings h , i : U × V −→ A. Suppose given such a pairing. The corresponding element φA ∈ HomA (U ⊗B V, A) gives rise via (2.3) tomorphisms φB ∈ HomB (U ⊗B V, HomA (B, A)), χA ∈ HomB (V, HomA (U, A)), andχB ∈ HomB (V, HomB (U, HomA (B, A))).
For the remainder of this section we keep ourselves in the above situation, andin addition assume as a running hypothesis that B, U and V are all locally freeof finite rank as A-modules. The first result that we prove will give necessary andsufficient conditions for the morphism φB to be surjective, under the assumptionthat the pairing h , i is perfect.
Lemma 2.4. In the setting described above, suppose that the pairing h , i : U ×
V −→ A is a perfect B-bilinear pairing of locally free A-modules of finite rank(that is, suppose that the morphism χA is an isomorphism of B-modules). Thenthe morphism φB : U ⊗B V −→ HomA (B, A) is surjective if and only if U/m is afaithful B/m-module for each maximal ideal m of A.
Proof. Since B is finitely generated over A, the morphism φB is surjective if andonly if the corresponding morphism φB /m : U/m ⊗B/m V /m −→ HomA (B, A)/mobtained by tensoring φB through with A/m over A is surjective, for each maximal
ideal m of A.
Since B is locally free over A, reduction modulo m yields an isomorphism∼

HomA (B, A)/m −→ HomA/m (B/m, A/m).
Composing φB /m with this isomorphism we obtain a morphism φB/m : U/m ⊗B/m
V /m −→ HomA/m (B/m, A/m). This corresponds via the diagram (2.3) (with A
replaced by A/m, B replaced by B/m, U replaced by U/m and V replaced byV /m) to the morphism φA/m = φA /m : U/m ⊗B/m V /m −→ A/m obtained bytensoring φA through with A/m over A. Since U is locally free over A, there is∼similarly an isomorphism HomA (U, A)/m −→ HomA/m (U/m, A/m), and composing
this with the morphism χA /m : V /m −→ HomA (U, A)/m we obtain a morphismχA/m : V /m −→ HomA/m (U/m, A/m), which also corresponds via (2.3) (again with
A replaced by A/m, and so on) to the morphism φA/m . Since χA is assumed to be
an isomorphism, we see that the same is true of χA/m . Putting these observations
together with that of the preceding paragraph, we see that it suffices to prove thelemma with A replaced by A/m, B by B/m, U by U/m and V by V /m. Thus forthe rest of the proof we assume that A is a field.
∼
By assumption χA induces an isomorphism V −→ HomA (U, A) of B-modules,
and so it suffices to prove the lemma in the case when V = HomA (U, A) and φA
corresponds to the tautological pairing U × HomA (U, A) −→ A. Thus in addition
to assuming that A is a field, we assume that V = HomA (U, A) from now on.
Now φB : U ⊗B HomA (U, A) −→ HomA (B, A) is surjective if and only if the
A-dual morphism B −→ HomA (U ⊗B HomA (U, A), A) is injective. Composing thismorphism with the adjointness isomorphism (2.1) (taking V to be HomA (U, A) and
W to be A), we obtain a morphism(2.5)

B −→ HomB (HomA (U, A), HomA (U, A)),

16

MATTHEW EMERTON

which is thus injective precisely when φB is surjective. One immediately checks that
(2.5) is the canonical morphism describing the B-module structure on HomA (U, A),
and so it is injective precisely when HomA (U, A) is a faithful B-module. Since
HomA (U, A) is a faithful B-module precisely when U is a faithful B-module, thiscompletes the proof of the lemma. One example of a situation in which the faithfulness condition of lemma 2.4 holds
is the case when U is a locally free B-module of finite positive rank. The following
lemma analyzes the case when A is a discrete valuation ring, and U is assumed to
be locally free as a B-module only over the generic point of Spec A.
Lemma 2.6. In addition to our running hypotheses, suppose that A is a discrete
valuation ring with uniformizer π and field of fractions K, and that U ⊗A K is freeof finite rank as a B ⊗A K-module. Then U/π is a faithful B/π-module if and only
if the natural map from B to the centre of the endomorphism ring EndB (U, U ) is
an isomorphism.
Proof. We begin with some remarks concerning the structure of EndA (U, U ) andits subring EndB (U, U ). Since U is free of finite rank over A, the former ring isa matrix ring over A. In particular it is torsion-free and its reduction modulo πacts faithfully on the reduction U/π. More generally, if R is any A-subalgebra of
EndA (U, U ), note that R/π acts faithfully on U/π if and only if R is saturated in
EndA (U, U ).
We let Z denote the centre of EndB (U, U ). Both Z and EndB (U, U ) are inparticular A-submodules of the matrix ring EndA (U, U ), and so are free of finite
rank as A-modules. Also, they are both saturated in EndA (U, U ), since U is torsionfree.
The action of B on U induces a morphism ι : B −→ Z (the natural map referred
to in the statement of the lemma). Tensoring ι with K over A yields the analogous
morphism of B ⊗A K into the centre of EndB⊗A K (U ⊗A K, U ⊗A K). By assumption
U ⊗A K is a free B ⊗A K-module of finite rank, and so its endomorphism ring is amatrix ring over B. Since the centre of such a ring is just B itself, we conclude thatι becomes an isomorphism after tensoring with K over A. Furthermore, the source
and target of ι are both free A-modules of finite rank, the source by assumption
and the target by the observations of the preceding paragraph. Thus we conclude
that ι is an injection, with torsion cokernel.
With these preliminaries complete, it is easy to prove the lemma. The morphism
ι is an isomorphism if and only if its cokernel is torsion free (since this cokernel
would then be both torsion and torsion-free, and so trivial). This cokernel is torsion
free if and only if ι(B) is saturated in Z. Since Z is saturated in EndA (U, U ), we
see that ι is an isomorphism if and only if ι(B) is saturated in EndA (U, U ). The
lemma follows once one recalls that ι(B) is saturated in EndA (U, U ) if and only if
B/π acts faithfully on U/π. 
The rest of this section is devoted to presenting criteria for computing the cokernel of φB , in situations where it may not be surjective (for example, because thepairing giving rise to φA is not perfect).
We add to our running hypotheses the assumption that A is a complete local
ring. Since B is locally free of finite rank as an A-module, this implies that B
decomposes as the direct sum of completions Bn , where n ranges over the (finitely
many) maximal ideals of B. For any such maximal ideal n, we let Un and Vn denote

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 17

the completions of U and V at n. Then U and V also decompose as the direct sumsof their completions.
Lemma 2.7. Suppose, in addition to our running hypotheses, that for any maximal
ideal n of B at least one of the following two conditions holds:(i) the morphisms Un ⊗Bn Vn −→ HomA (Bn , A) induced by φB and Vn −→HomA (Un , A) induced by χA are both surjective;(ii) the Bn -module Un is free of rank one.
Then the cokernel of the morphism φB is isomorphic to the cokernel of the morphism χA .
∼∼∼Proof. Writing B −→ ⊕n Bn , U −→ ⊕n Un , and V −→ ⊕n Vn , we see that the cokernels of φB and χA also decompose as such direct sums, and so it suffices to verify
the lemma after replacing B by one of the Bn and U and V by the corresponding
Un and Vn . Thus we may assume that either φB and χA are surjections, or that U
is a free B-module of rank one.
In the first case, both φB and χA have trivial cokernels, and so the lemma holds
in this case.
In the second case, note that the morphism χB is equal to the composition of χA∼with the isomorphism HomA (U, A) −→ HomB (U, HomA (B, A)). Thus the cokernelof χA is isomorphic to the cokernel of χB . In addition, the morphism χB can bethought of as being obtained from the morphism φB by tensoring over B with theB-dual of the B-module U (which we are assuming to be free of rank one). Thus
the cokernel of φB is isomorphic to the cokernel of χB , and so to the cokernel ofχA , proving the lemma in this case. For ease of future reference, we recall our notation and running hypotheses. We
are given a complete local ring A, an A-algebra B which is free of finite rank as an Amodule, and a pair of B-modules U and V which are free of finite rank as A-modules.
Furthermore, there is a B-bilinear pairing U × V → A, which induces tautologicalmaps of B-modules χA : V → HomA (U, A) and φB : U ⊗B V → HomA (B, A).
Proposition 2.8. Suppose, in addition to our running hypotheses, that for any
maximal ideal n of B at least one of the following two conditions holds:(i) the pairing Un × Vn −→ A is perfect, and Un /m is a faithful A/m-module(here m denotes the maximal ideal of A);(ii) the Bn -module Un is free of rank one.
Then the cokernel of the morphism φB is isomorphic to the cokernel of the morphism χA .
Proof. The pairing Un × Vn −→ A is perfect if and only if the morphism Vn −→HomA (Un , A) induced by χA is an isomorphism. Thus lemma 2.4 (with U , V and
B replaced by Un , Vn and Bn ) implies that condition (i) of the present proposition
implies condition (i) of lemma 2.7. Since condition (ii) of the present proposition
is identical to that of lemma 2.7, the proposition is now seen to follow from thatlemma. 
3. The T-module X . Constructions.
Let S denote the set of singular points of the geometric fibre of X0 (p) in characteristic p. As in section 0 we write S = {x0 , . . . , xg } (where g is equal to thegenus of X0 (p)). The set S is in bijection with the set of isomorphism classes of

18

MATTHEW EMERTON

supersingular elliptic curves over Fp , and for each point xi ∈ S we let Ei denote a
supersingular curve representing the isomorphism class corresponding to xi .
We let X denote the free Z-module on the set S, let deg : X −→ Z denote the Zlinear map obtained by sending each xi ∈ S to 1 ∈ Z, and let X 0 denote the kernelof deg. The Hecke correspondences on X0 (p) induce endomorphisms of X and of
X 0 . (See [20, pp. 443–445] for a detailed discussion of these correspondences.)
Theorem 3.1. (i) The action of the Hecke correspondences on Z, regarded as
the quotient X /X 0 , factors through TEis . That is, the Hecke operator Tn acts as
multiplication by σ 0 (n).
(ii) The action of the Hecke correspondences on X 0 makes X 0 a faithful T0 module.
(iii) The action of the Hecke correspondences on X makes X a faithful T-module.
Proof. Let T0 denote the polynomial ring over Z on the countably many generators
Tn . The action of the Hecke correspondences on X induces an action of T0 on X .
Let I 0 denote the kernel of the surjection of T0 onto TEis , and let J 0 denote thekernel of the surjection of T0 onto T0 . Then I 0 ∩ J 0 is the kernel of the surjection
T0 −→ T, since the map T −→ TEis ⊕ T0 is injective. Furthermore, since thecokernel of this map is finite of order n, we see that nT0 ⊂ I 0 + J 0 , and hence thatn(I 0 ∩ J 0 ) ⊂ I 0 J 0 .
If we prove (i) and (ii), then we conclude immediately that AnnT0 (X ) ⊂ I 0 ∩ J 0 ; aconsideration of the exact sequence 0 −→ X 0 −→ X −→ Z −→ 0 shows in addition
that I 0 J 0 ⊂ AnnT0 (X ). Taking into account the inclusion n(I 0 ∩ J 0 ) ⊂ I 0 J 0 and thefact that X is Z-torsion free, we deduce that I 0 ∩ J 0 = AnnT0 (X ). This is the claimof (iii).
For any xi ∈ S, deg(Tn xi ) = σ 0 (n). Thus the action of T0 on Z (as it appears in∼
the preceding exact sequence) kills I 0 , and yields a faithful action of T0 /I 0 −→ TEison Z. This proves (i).
Let T0 (p) denote the torus that is the connected component of the fibre in characteristic p of the Néron model of the Jacobian J0 (p) of X0 (p). The Hecke correspondences act on T0 (p) functorially via their action on J0 (p), and hence also act
on the character lattice Hom(T0 (p)/Fp , Gm/Fp ) of T0 (p). Furthermore (a) there isa canonical isomorphism (of Hecke modules) of X 0 with this character lattice; (b)we can recover the Hecke action on T0 (p) from the Hecke action on its characterlattice; (c) the endomorphisms of J0 (p) act faithfully on its toric reduction T0 (p);(d) we can recover the Hecke action on cusp forms from the Hecke action on J0 (p)by passing to its cotangent space. Thus the annihilator of the T0 -action on X 0 is
equal to annihilator of the T0 -action on the space of weight two cusp forms on Γ0 (p),which is J 0 . This proves (ii). (This is a particular case of [20, thm. 3.10].) 
Remark. A more natural way to prove part (iii) of this theorem, without studying
separately the Eisenstein and cuspidal parts of X , would be to observe that X is thecharacter group of the connected component of the fibre in characteristic p of theNéron model of the generalized Jacobian of the non-proper curve Y0 (p) = X0 (p) \cusps, and that the Hecke action on this semi-abelian variety factors faithfully
through T.
Corollary 3.2. X ⊗Z Q is a free T ⊗Z Q-module of rank one.
Proof. It follows from theorem 3.1 that X ⊗Z Q is a faithful T ⊗Z Q-module. Also
T ⊗Z Q is a semi-simple Q-algebra. (This is proved by noting that T ⊗Z R acts

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 19

faithfully as an algebra of self-adjoint operators on a finite-dimensional vector spaceequipped with a positive definite inner product. One can use the vector space M ⊗Z
R equipped with the Petersson inner product, as in [1], or the space X ⊗Z R equippedwith the inner product induced by definition 3.4 below, as in [6, prop. 2.7 (7)].)
Furthermore, T ⊗Z Q and X ⊗Z Q are of the same dimension (= g + 1) over Q.
These observations suffice to prove the corollary. For each xi ∈ S we let ei denote one half of the number of automorphisms of
Ei . The following lemma is pointed out in [6, p. 117]. (Recall that ∆ denotes thedenominator of (p − 1)/12 when this fraction is written in lowest terms.)
Lemma 3.3. The productgYei is equal to ∆.

i=0

Proof. This follows from the well-known explicit calculation of the ei , which we recall. If p = 2 (respectively p = 3) then j = 0 is the unique supersingular j-invariant,and the corresponding elliptic curves has 24 (respectively 12) automorphisms. If(p, 6) = 1 then ei = 1 unless j(Ei ) = 0 (respectively j(Ei ) = 1728), in which caseei = 3 (respectively ei = 2); furthermore, this j-value actually is supersingular incharacteristic p if and only if p ≡ −1 (mod 3) (respectively p ≡ −1 (mod 4)). 
Definition 3.4. We define a symmetric pairing on X via the formulahxi , xj i = ei δi,j = ej δi,j = one-half the number of isomorphisms from Ei to Ej .
If we restrict this pairing to the submodule X 0 of X , we obtain the PicardLefschetz pairing discussed in [20, §’s 2 and 3].
We now prove a series of results relating this pairing to the theta series of quadratic spaces constructed from the supersingular elliptic curves Ei . There is someoverlap with the results of [6]; in particular, proposition 3.6, lemmas 3.7 and 3.8,
and corollary 3.9 are essentially contained in sections 2, 4 and 5 of this reference,
although the point of view is different. (In [6], the lattice X is considered as thePicard group of a certain curve obtained as the disjoint union of genus zero curves.
Regarding notation, note that in [6] this curve is denoted by X, while the lattice
we denote by X is denoted by Pic(X). It might also be useful to observe that thelattice we denote by X is denoted by Λ in [20], and that the lattice we denote by
X 0 is denoted by X in that reference.)
We begin by recalling a construction from [6] and [8]:Definition 3.5. For each pair of elements xi and xj of S, let Li,j denote thequadratic space whose underlying Z-module is the space of isogenies from Ei to Ej ,equipped with the quadratic form φ 7→ deg(φ). (Here and below we adopt the usual
convention that if φ = 0 then deg(φ) = 0.)
Proposition 3.6. For any two elements xi and xj of S, Li,j is a free Z-module ofrank 4 equipped with a positive definite quadratic form.
Proof. Since any two supersingular elliptic curves in characteristic p are isogenous,
we see that the rank of Li,j as a Z-module is equal to the rank of Li,i for any choiceof i, and this rank is 4, since the endomorphisms of a supersingular elliptic curve
form an order in a quaternion algebra over Q. That taking degrees yields a definite
quadratic form is immediate from the fact that an isogeny of degree zero must be
the zero isogeny. 

20

MATTHEW EMERTON

Lemma 3.7. For any two elements xi and xj of S there is a natural isomorphism∼of quadratic spaces Li,j −→ Lj,i .
Proof. The required isomorphism is obtained by sending any isogeny from Ei to
Ej , of some degree n say, to its dual isogeny, which maps from Ej to Ei , and is alsoof degree n. 
If n is any natural number, let rn (Li,j ) denote the number of elements of Li,j ofdegree n.
Lemma 3.8. If xi and xj are any two elements of S, thenhTn xi , xj i = one-half the number of degree n isogenies from Ei to Ej
1= rn (Li,j ).
2
Proof. Since Tn xi is the formal sum of all curves that are the image of Ei under
an n-isogeny, and since hxk , xj i equals one half the number of isomorphisms from
Ek to Ej , for any index k, we see that hTn xi , xj i does indeed equal one-half thenumber of degree n isogenies from Ei to Ej , as claimed. 
Corollary 3.9. The pairing of definition 3.4 is T-bilinear. That is, for any x, y ∈X and any T ∈ T, hT x, yi = hx, T yi.
Proof. It suffices to verify this when x = xi and y = yj are elements of S and
T = Tn for some natural number n. In this case lemmas 3.7 and 3.8 show that11
hTn xi , xj i = rn (Li,j ) = rn (Lj,i ) = hTn xj , xi i = hxi , Tn xj i. 
22Corollary 3.9 shows that we are in the situation of section 2 (taking A to be Z,
B to be T and U = V to be X ). In particular the pairing of definition 3.4 yields
a morphism of Z-modules X ⊗T X −→ Z (the morphism denoted φA in section 2)
and a corresponding morphism of T-modules(3.10)X ⊗T X −→ HomZ (T, Z)(the morphism denoted φB in section 2). Composing (3.10) with the isomorphismof proposition 1.3 (ii) we obtain a morphism θ : X ⊗T X −→ N . From the definitionof the pairing of proposition 1.3 and the construction of (3.10) we see that the mapθ is determined by the formula an (θ(x ⊗ x0 )) = hTn x, x0 i, for x, x0 ∈ X and n ≥ 1.
The following results are devoted to computing the map θ, and in particular, will
explain our choice of notation for this map.
Proposition 3.11. If xi and xj are any two elements of of S, then for any n ≥ 1,
1
we have an (θ(xi ⊗ xj )) = rn (Li,j ).
2
Proof. Combining the above discussion with lemma 3.8 we see that if n ≥ 1 thenan (θ(xi ⊗ xj )) = hTn xi , xj i = 21 rn (Li,j ). 

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 21

Definition 3.12. Let x denote the element of X defined by x = ∆

g
X
xi
i=0

ei

. (Notethat lemma 3.3 implies that this is an element of X .)This is the element denoted by e0 in [6, (4.8)]. Observe that hx, xi i = ∆ for eachelement xi of S. Thus hx, xi = ∆ deg(x), for any x ∈ X .
Let X Eis denote the maximal T-submodule of X on which the action of T factors
through TEis . The short exact sequence 0 −→ X 0 −→ X −→ Z −→ 0 along withparts (i) and (ii) of theorem 3.1 shows that X Eis must be a rank one Z-submoduleof X .
Lemma 3.13. X Eis is the free Z-module of rank one generated by x.
Proof. For any element xj of S, we compute thathTn x, xj i = hx, Tn xj i = ∆X hxi , Tn xj i
i

=∆

X

ei

number of n-isogenies with source Ej and target isomorphic to Eii= ∆ × number of n-isogenies with source Ej = ∆σ 0 (n).
Thus the coefficient of xj in Tn x must be ∆σ 0 (n)/ej . Since j was arbitrary, we findthat Tn x = σ 0 (n)x. Thus x lies in X Eis .
Now X Eis is free of rank one over Z, and the coefficients ∆/ei of the generators
xi of X in x are coprime elements of Z (as follows from lemma 3.3 and the explicit
values of the numbers ei which are recalled in the course of its proof). Thus we seethat x is not divisible by any non-unit element of Z in X , and so it must generate
X Eis . 
Proposition 3.14. Let xi be any element of S. Then θ(x ⊗ xi ) is equal to themodular form ∆E.
Proof. Similarly to the proof of proposition 3.11, we computean (θ(x ⊗ xi )) = hTn x, xi i = hσ 0 (n)x, xi i = ∆σ 0 (n),since hx, xi i = ∆ for each generator xi of X . 
Proposition 3.15. The following statements hold:1;2(ii) for any two elements x, y ∈ X , the constant term of θ(x ⊗ y) is equal to(deg(x) deg(y))/2;
1(iii) if xi and xj are any two elements of S, then θ(xi ⊗ xj ) = Θ(Li,j );2
X 1(iv) a0 (E) =;2eii(v) deg x = n;
X 1
p−1
(vi)
=
.
ei
12
i
(i) if xi and xj are any two elements of S, then a0 (θ(xi ⊗ xj )) =Proof. We begin by proving that the six statements are mutually equivalent. Since
the formation of a0 (θ(x ⊗ y)) and (deg(x) deg(y))/2 are both bilinear in x and y,

22

MATTHEW EMERTON

statements (i) and (ii) are clearly equivalent. Proposition 3.11 together with thefact that a0 (Θ(Li,j )) = 1 shows that statements (i) and (iii) are equivalent. Since
X 1deg(x) = ∆and a0 (E) = (p − 1)/24 = n/2∆, we see that statements (iv), (v)eiiand (vi) are equivalent.
We now show that statements (i) and (iv) are equivalent. To this end, we observethat
a0 (θ(xi ⊗ xj )) =1
(a0 (θ(x ⊗ xj )) + a0 (θ((deg(x)xi − x) ⊗ xj ))) .
deg(x)

Note that deg(x)xi − x lies in X 0 , which implies that θ((deg(x)xi − x) ⊗ xj ) lies in
M0 , and so has vanishing constant term. Proposition 3.14 asserts that θ(x ⊗ xi ) =∆E. Thus we find that
X 1
deg(x)a0 (θ(xi ⊗ xj ))a0 (E) == a0 (θ(xi ⊗ xj )),∆eiiwhich makes it clear that statements (i) and (iv) are equivalent.
It remains to see that these statements actually hold. There are at least two
ways to do this. For example, it is a classical theorem that the theta-series Θ(Li,j )are weight two modular forms on Γ0 (p). The formula of proposition 3.11 then shows
1that θ(xi ⊗ xj ) and Θ(Li,j ) coincide up to their constant terms, and so coincide.
2
1
1
Thus a0 (θ(xi ⊗ xj )) = a0 (Θ(Li,j )) = , proving part (i) of proposition 3.15.
2
2
Alternatively, one could observe that the condition (vi) is precisely Eichler’s mass
formula [4]. 
Propositions 3.11 and 3.15 “almost” give an analysis-free proof of the modularityof the theta series Θ(Li,j ). Indeed, the construction of the map θ is purely algebrogeometric, and proposition 3.11 then shows that Θ(Li,j ) is a modular form, provided
we allow ourselves to modify its constant term as necessary. Proposition 3.15 showsthat in fact no such modification is necessary. To what extent does the proof ofthis theorem depend on analysis? Igusa [13] has given a proof of Eichler’s mass
formula that exploits the properties of supersingular elliptic curves and makes noreference to analytic considerations. If one appeals to this argument, then the only
analytic ingredient in the proof of proposition 3.15 is the theory of the Eisensteinseries E. One can easily give a purely algebro-geometric proof that there exists a
Hecke eigenform E ∈ N on which T acts through TEis , but one is left with the
problem of determining a0 (E), and the usual proof that this is equal to (p − 1)/24
uses analysis. It would be interesting to have a purely geometric determination ofthis constant term, for example in the form of statement (iv) of proposition 3.15.
This would then yield a purely algebro-geometric determination of the constantterm of the modular forms θ(xi ⊗ xj ). (For additional remarks on this problem, see
the discussion following the proof of proposition A.2 of the appendix.)
Lemma 3.16. (i) The cokernel of the morphism X −→ HomZ (X , Z) induced by
the pairing of definition 3.4 is isomorphic to Z/∆, regarded as a T-module via thesurjection T −→ TEis . In particular the discriminant of this pairing is equal to ∆.
(ii) The cokernel of the morphism X 0 −→ HomZ (X 0 , Z) induced by the restriction of the pairing of definition 3.4 to X 0 is isomorphic to Z/n, regarded as a

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 23

T-module via the surjection T −→ TEis . In particular the discriminant of thispairing is equal to n.
Proof. The claim of part (i) concerning the discriminant of the pairing of definition
3.4 follows directly from the fact that with respect to the basis S of X , this pairing
is given by a diagonal matrix, the product of whose diagonal entries equals ∆, bylemma 3.3.
Since the pairing of definition 3.4 has a non-zero discriminant, it induces anembedding X −→ HomZ (X , Z), which is T-linear, by corollary 3.9. Since hx, xi =∆ deg(x) for all x ∈ X , we see that under this map the element x of X maps
to the element ∆ deg of HomZ (X , Z). Thus we obtain a commutative diagram ofmorphisms of T-modules(3.17)

0/ X Eis(1)0
/ Z deg/ X Eis ⊕ X 0

X

/ X0

/0

(2a)
(3)
(2b)


/ HomZ (X , Z)
/ HomZ (X 0 , Z)

/ 0,

in which the top and bottom rows are exact, and in which all vertical arrows are
injections.
Map (1) of (3.17) has cokernel equal to Z/∆, regarded as a T-module via thesurjection T −→ TEis (as follows from part (i) of theorem 3.1), (2a) has cokernel
isomorphic to Z/n, regarded as a T-module via the surjection T −→ TEis (as follows
from part (i) of theorem 3.1 and part (v) of proposition 3.15), and (2b) has cokernelof order ∆ (as follows from the discriminant calculation made at the beginning of
this proof). Since ∆ and n are coprime, we deduce that the cokernel of (1) maps
isomorphically to the cokernel of (2b), proving part (i), while the cokernel of (2a)maps isomorphically to the cokernel of (3), proving part (ii). 
We remark that part (ii) of lemma 3.16 can also be proved in the following way:
it follows from [9, 11.5.2b] that the cokernel of the map X 0 −→ HomZ (X 0 , Z) is
isomorphic to the group of connected components of the geometric fibre at p of theNéron model of J0 (p), which is a cyclic group of order n on which T acts throughits quotient TEis , by [15, thm. A.1, prop. II.11.1]. Note though that this argument
also depends on calculating the discriminant of the Picard-Lefschetz pairing, which
in turn depends on Eichler’s mass formula (see proposition 1.4 and section 2, as
well as the remark prior to the statement of proposition 1.2, of [15, appendix]), andso this proof is not so different from the one that we have given. As a final remark
on lemma 3.16, we also note that theorem 3.12, proposition 3.14 and theorem 3.22of [20] each imply the weaker statement that the cokernel of the map from X 0 into
Hom(X 0 , Z) is annihilated by the Eisenstein ideal, and so in particular by n.
Let ` be a prime. The tensor product T ⊗Z Z` is a direct sum of the complete
local rings Tm , where m ranges over the maximal ideals of T of residue characteristic`. Thus the tensor product X ⊗Z Z` decomposes as a direct sum of its completions
Xm . We may tensor the pairing of definition 3.4 by Z` over Z to obtain a pairing(3.18)X ⊗Z Z` × X ⊗Z Z` −→ Z` .

24

MATTHEW EMERTON

Since this pairing is T ⊗Z Z` -bilinear (by corollary 3.9) we see that for distinct
maximal ideals m and m0 of residue characteristic `, this pairing becomes trivial
when restricted to Xm × Xm0 . Thus this pairing is the direct sum of its restrictions Xm × Xm −→ Z` , as m ranges over the maximal ideals of T having residue
characteristic `.
Similarly, we can restrict (3.18) to obtain the pairing(3.19)

X 0 ⊗Z Z` × X 0 ⊗Z Z` −→ Z` ,and we similarly observe that it decomposes as the direct sum of its restrictions
Xm0 × Xm0 −→ Z` .
Corollary 3.20. Let m be a maximal ideal of T of residue characteristic `.
(i) If ` does not divide ∆ then the pairing Xm ×Xm −→ Z` is a perfect Tm -bilinear
pairing of free Z` -modules.
(ii) If m is not an Eisenstein prime, then the pairing Xm0 ×Xm0 −→ Z` is a perfect0Tm -bilinear pairing of free Z` -modules.
Proof. Part (i) follows immediately from the discriminant computation of part (i)of lemma 3.16.
To prove part (ii), we proceed by cases. If ` does not divide ∆, then part (ii)∼follows from part (i) and the direct sum decomposition XmEis ⊕ Xm0 −→ Xm , together
with the fact that X Eis and X 0 are orthogonal under the pairing (by T-bilinearity).
On the other hand, if ` does not divide n, then part (ii) follows from the discriminantcalculation of lemma 3.16 (ii). Since ∆ and n are coprime by construction, we seethat part (ii) is now proved. 
Let us point out that part (ii) of corollary 3.20 may also be deduced from[20, thm. 3.12/prop. 3.14/thm. 3.22]. (Compare the related remark following theproof of lemma 3.16.)
The final observation that we will make in this section is that the results alreadyestablished suffice to recover Eichler’s theorem [5] that the space of weight two
modular forms on Γ0 (p) is spanned by theta series.
Proposition 3.21. The morphism X ⊗Z Q ⊗T⊗Z Q X ⊗Z Q −→ M ⊗Z Q inducedby (0.2) is an isomorphism. In particular, the Q-vector space M ⊗Z Q is spannedby the theta series Θ(Li,j ).
Proof. It follows from lemma 3.16 (i) that the pairing of definition 3.4 has non-zero
discriminant. Thus if we take A = Q, B = T ⊗Z Q, and U = V = X ⊗Z Q, wefind that it induces a perfect pairing U × U −→ A, and so we are in the situationof lemma 2.4. It follows from that lemma, together with corollary 3.2, that themorphism φB (in the notation of that lemma) is an isomorphism. However, φB
is precisely the morphism θ tensored with Q over Z (once we use propositions 1.1∼∼
and 1.3 to make the identifications M⊗Z Q −→ N ⊗Z Q −→ HomQ (T⊗Z Q, Q)). The
proposition now follows by observing that proposition 3.11 implies that θ tensoredwith Q over Z and (0.2) tensored with Q over Z are the same map. 
4. The T-module X . Further properties.
In this section, we study in more detail the structure of the Tm -modules Xm and
Xm0 (as m ranges over the maximal ideals of T), and their relation to the arithmeticof the Jacobian J0 (p).

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 25

We begin by studying the case when m is an Eisenstein maximal ideal of T of
residue characteristic `. As in the proof of theorem 1.10, we let I denote the kernel∼
0of the map Tm −→ TEism −→ Z` and J denote the kernel of the map Tm −→ Tm .
Then I and J are each principal (since their intersection is zero and each inducesa Cartier divisor in the closed subscheme cut out by the other – see the proofs oftheorem 1.10 and lemma 1.11), and so I + J is generated by a pair of elementsη ∈ I, γ ∈ J. Now γ is a lifting of the generator of the image of J in Z` , and so iscongruent to some power `n of ` modulo I. Thus we have I + J = (η, `n )Tm , and∼Tm /(I + J) −→ Z` /`n . We conclude that m = (η, `)Tm .
Lemma 4.1. Let m be an Eisenstein maximal ideal of residue characteristic `.
(i) If N is any finitely generated Tm /`-module, then N [m] (the T-submodule ofm-torsion points of N ) and N/m are T/m-vector spaces of equal rank.
(ii) If 0 −→ N 0 −→ N −→ N 00 −→ 0 is a short exact sequence of Tm /`-modules,
there is an induced long exact sequence
0 −→ N 0 [m] −→ N [m] −→ N 00 [m] −→ N 0 /m −→ N/m −→ N 00 /m −→ 0.
Proof. To prove (i), note that N [η] and N/η are Tm -modules annihilated by both` and η, and hence by m, so that N [η] = N [m] and N/η = N/m. That they areof equal rank as T/m-vector spaces (equivalently, of equal length as Tm -modules)·η
follows from the exact sequence 0 −→ N [η] −→ N −→ N −→ N/η −→ 0 and the
additivity of lengths in exact sequences.
·η
To prove (ii), tensor the given exact sequence with the complex Tm /` −→ Tm /` offree Tm /`-modules and apply the snake lemma. This yields the long exact sequence
0 −→ N 0 [η] −→ N [η] −→ N 00 [η] −→ N 0 /η −→ N/η −→ N 00 /η −→ 0.
Since ` and η generate the maximal ideal mTm , we can rewrite this as
0 −→ N 0 [m] −→ N [m] −→ N 00 [m] −→ N 0 /m −→ N/m −→ N 00 /m −→ 0,
as claimed in the statement of part (ii). 
Theorem 4.2. Let m be an Eisenstein maximal ideal of T. Then Xm is a freeTm -module of rank one, and Xm0 is a free T0m -module of rank one.
Proof. As above, we let ` denote the residue characteristic of m. From corollary 3.2
we see that Xm ⊗Z` Q` is free of rank one over Tm ⊗Z` Q` , and that similarly Xm0 ⊗Z` Q`
is free of rank one over T0m ⊗Z` Q` . Thus to prove the theorem it suffices to show∼that each of the T/m −→ T0 /m-vector spaces X /m and X 0 /m are of dimension
one.
We begin by proving that these two vector spaces are of the same dimension. To
do this, we first tensor the short exact sequence of T-modulesdeg

0 −→ X 0 −→ X −→ Z −→ 0with the flat T-module Tm to obtain the short exact sequence of Tm -modulesdeg

0 −→ Xm0 −→ Xm −→ Z` −→ 0.

26

MATTHEW EMERTON

This is a short exact sequence of torsion-free (equals flat) Z` -modules, and so reducdeg

ing modulo ` yields the short exact sequence 0 −→ Xm0 /` −→ Xm /` −→ Z/` −→ 0,
giving rise, by part (ii) of lemma 4.1, to the long exact sequencedeg(4.3) 0 −→ Xm0 /` [m] −→ Xm /` [m] −→ Z/`
deg−→ X 0 /m −→ X /m −→ Z/` −→ 0.
∼∼(Note that T acts on the quotient Z −→ X /X 0 through TEis , so that Z/` [m] −→∼(Z/`)/m −→ Z/`.) We wish to show that X /m and X 0 /m are of the same dimension
over T/m. It is equivalent to show (by lemma 4.1, part (i)) that X /` [m] andX 0 /` [m] are of the same rank. A consideration of (4.3) shows that it suffices toprove that the image of X /` [m] in Z/` is trivial.
Let x̄ ∈ Xm /` [m]. We can find an element x ∈ X that reduces to x̄ modulo `.
Let xi be an element of S. Write f = θ(x ⊗ xi ) ∈ M. Then f − a1 (f )E is anelement of `N , since x is annihilated by m modulo `, and so (in the notation of
proposition 1.1) A0 (f ) ≡ A0 (a1 (f )E) = a1 (f )n mod `. By proposition 3.15 we seethat A0 (f ) = 2∆ deg(x)/2 = ∆ deg(x). Thus we conclude that ∆ deg(x) ≡ a1 (f )nmod `. Since m is an Eisenstein prime, proposition 1.7 shows that ` divides n, andso is coprime to ∆. Thus deg(x) ≡ 0 mod `, and so x̄ does have trivial image inZ/`. This completes the proof that X /m and X 0 /m have the same dimension.
To prove the theorem, it remains to show that X 0 /m is of dimension one over0
T /m. We will follow the methods of [20] and [23], and we refer to the proofs of[20, thm. 6.1] and [23, thm. 2.3] and the references cited therein for a more detailed
discussion of the techniques used and claims made in the argument that follows.
Let J denote the Néron model of the Jacobian J0 (p) of X0 (p), and let T0 (p)denote the connected component of the characteristic p fibre of J . Then T0 (p)is a torus over Fp . Recall that X 0 can be naturally identified with the characterlattice HomFp (T0 (p), Gm ) of T0 (p). Since T0 (p) is defined over Fp , we see that X 0
is equipped with an action of Gal(Fp /Fp ), and that T0 (p) can in turn be recoveredfrom X 0 equipped with its action of Gal(Fp /Fp ) as Hom(X 0 , Gm ) (at first definedover Fp , and then descended to Fp using the descent data induced by the Galois
action on X 0 ). By regarding the Gal(Fp /Fp )-action on X 0 as being an unramified
action of Gal(Qp /Qp ), we see that the expression Hom(X 0 , Gm ) equally well definesa torus T over Zp that has T0 (p) as its special fibre. (We are recalling the standard
fact that any torus over the special fibre of a Henselian discrete valuation ring
deforms uniquely to a torus over the discrete valuation ring itself.)
The formal completion of T along its characteristic p fibre embeds into theformal completion of J along its characteristic p fibre, and so the group scheme
T [`] embeds as a sub-group scheme of J [`] (at first formally, but then actually,
since it is finite). Passing to characteristic zero fibres shows that T [`]/Qp embeds
into J0 (p)[`]/Qp . Now T [`] is Cartier dual to X 0 /`, and the cokernel of the precedingembedding is naturally identified with X 0 /`. We thus obtain a short exact sequenceof Gal(Qp /Qp )-modules
0 −→ Hom(X 0 /`, µ` ) −→ J0 (p)[`](Qp ) −→ X 0 /` −→ 0,
where Hom(X 0 /`, µ` ) is naturally identified with T [`](Qp ). (This is the short exact
sequence [23, (2.4)].)

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 27

Apply HomT0 (T0 /m, –) to this short exact sequence, to obtain the exact sequence
0 −→ Hom(X 0 /m, µ` ) −→ J0 (p)[m](Qp ) −→ X 0 /` [m],where J0 (p)[m] denotes the finite subgroup scheme of J0 (p)[`] that is the simultaneous kernel of all element of m. Observe that [15, p. 99 and cor. II.16.3] imply
that J0 (p)[m] is a two dimensional T/m-vector space scheme, and that J [m] extends it to a two-dimensional finite flat T/m-vector space scheme over Zp , whose
specialization to characteristic p has non-zero image in the connected component
group of the special fibre of J . On the other hand, it follows from the discussionof the preceding paragraph that the specialization to characteristic p of the pointsof Hom(X 0 /m, µ` ) lie in T0 (p)[`], and so have trivial image in this connected component group. Thus Hom(X 0 /m, µ` ) must be only one-dimensional over T/m. The
theorem now follows. 
We now turn to an analysis of Xm and Xm0 when m is of the first type (so that∼∼Tm −→ T0m and Xm0 −→ Xm .)Lemma 4.4. Let m be a maximal ideal of T of the first type, having residue characteristic `, and let W be a finite length T0m /`-module. Then the T0m /`-modules
Xm0 /` ⊗T0m /` W and HomT0m /` (W, Xm0 /`) are naturally Z/`-dual to one another. In
particular, they are of equal dimension as Z/`-vector spaces.
Proof. Since m is non-Eisenstein, it follows from corollary 3.20 (ii) that the pairing(3.19) induces a perfect T0m /`-bilinear pairing Xm0 /` × Xm0 /` −→ Z/`. Using this wecompute (using the adjointness between Hom and ⊗) that∼

HomZ/` (Xm0 /` ⊗T0m /` W, Z/`) −→ HomT0m /` (W, HomZ/` (Xm0 /`, Z` ))∼−→ HomT0m /` (W, Xm0 /`).
Since a Z/`-vector space and its dual have the same dimension, this proves thelemma. 
For the remainder of this section we fix an embedding Q → Qp and thus obtainan embedding of Galois groups Gal(Qp /Qp ) → Gal(Q/Q). (In other words, we fix
a choice of decomposition group at p in Gal(Q/Q).) The following simple lemma
will be required in the proof of theorem 4.6 below. We refer to [24, p. 189] for thenotion of finiteness of a Gal(Qp /Qp )-representation.
Lemma 4.5. Let A be a finite local ring with residue field k, let ρ : Gal(Q/Q) →GL2 (k) be an irreducible two-dimensional representation of Gal(Q/Q) over k, let
U be a finite-length A-module equipped with an action of Gal(Q/Q), and let V be a
finite-length A-module equipped with an action of Gal(Qp /Qp ). Suppose that(i) ρ is not finite at p;(ii) the semi-simplification of U as an A[Gal(Q/Q)]-module is isomorphic to a
direct sum of copies of ρ;(iii) V is a finite Gal(Qp /Qp )-module;(iv) there is a Gal(Qp /Qp )-equivariant A-linear embedding V → U .
Then lg(U ) ≥ 2 lg(V ). (Here lg(–) denotes the length of a finite length Amodule.)
Proof. Let F 0 ⊃ F 1 ⊃ · · · ⊃ F l−1 ⊃ F l = 0 be a decreasing filtration of U by
A[Gal(Q/Q)]-modules such that the successive quotients F i /F i+1 are isomorphic

28

MATTHEW EMERTON

to ρ (i = 0, . . . , l − 1). Let Gi = V ∩ F i . Then Gi /Gi+1 embeds into F i /F i+1 .
Since the latter k-vector space is two-dimensional and not finite as a Gal(Qp /Qp )module, while Gi /Gi+1 is finite (being a subquotient of the finite representation V ),Pl−1
we see that Gi /Gi+1 is at most one-dimensional. Thus lg U = i=0 lg F i /F i+1 ≥Pl−12 i=0 lg Gi /Gi+1 = 2 lg V. 
If m is a maximal ideal of the first type then attached to m there is a twodimensional irreducible representation ρm : Gal(Q/Q) −→ GL2 (T0 /m), characterized by the property that it is unramified away from p and `, and that for any
prime q distinct from p and `, the characteristic polynomial of ρ(Frobq ) is equal
to X 2 − Tq X + q (mod m) [15, prop. II.14.1]. Furthermore, the T/m [Gal(Q/Q)]module J0 [m](Q) has a Jordan-Hölder filtration whose successive irreducible quotients are isomorphic to the representation ρm [15, proof of prop. II.14.2].
Note that the set S that forms a basis for X is equipped with an action ofGal(Fp /Fp ) (since it is a set of Fp -valued points of a curve defined over Fp ). Thus X
is naturally a Gal(Fp /Fp )-module. We may equally well regard X as an unramified
Gal(Qp /Qp )-module. (This Galois action coincides with that considered in the
proof of theorem 4.2.)
Theorem 4.6. Let m be a maximal ideal of T of the first type of characteristic `.
Then there is a short exact sequence of T0m /` [Gal(Qp /Qp )]-modules
0 −→ Hom(Xm0 , µ` ) −→ J0 (p)[`](Qp )m −→ Xm0 /` −→ 0,
which is split as a short exact sequence of T0m /`-modules.
Proof. The theorem is an extension of [23, thm. 2.3], and will follow the same linesof proof as that result. Just as in the proof of theorem 4.2 we obtain a short exact
sequence
0 −→ Hom(X 0 /`, µ` ) −→ J0 (p)[`](Qp ) −→ X 0 /` −→ 0
of T0 /` [Gal(Qp /Qp )]-modules, which we may localize at m to yield the required
short exact sequence of T0m /` [Gal(Qp /Qp )]-modules(4.7)

0 −→ Hom(Xm0 /`, µ` ) −→ J0 (p)[`](Qp )m −→ Xm0 /` −→ 0.

We will show that (4.7) remains short exact after applying HomT0m /` (W, –), forany finite length T0m /`-module W . Taking W = Xm0 /` will then yield a T0m /`-linear
map s : Xm0 /` → J0 (p)[`](Qp )m which lifts the identity map from Xm0 /` to itself,and so splits (4.7).
Choose a finite length T0m /`-module W . Applying HomT0m /` (W, –) to (4.7) yields
an exact sequence of T0m /` [Gal(Qp /Qp )]-modules(4.8) 0 −→ Hom(Xm0 /` ⊗T0m /` W, µ` )−→ HomT0m /` (W, J0 (p)[`](Qp )m ) −→ HomT0m /` (W, Xm0 /`).
(Here we regard W as being endowed with the trivial Gal(Qp /Qp )-action. We∼
have also used the isomorphism HomT0m /` (W, Hom(Xm0 /`, µ` )) −→ Hom(Xm0 /`⊗T0m /`W, µ` ) provided by the adjointness between Hom and ⊗.) We must show that (4.8)

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 29

is exact on the right. We begin by making some observations concerning the Galoisaction on the first two terms of this exact sequence.
(i) The Gal(Qp /Qp )-module Hom(Xm0 /` ⊗T0m /` W, µ` ) is finite at p. (To see this,note that if we forget the T0m /`-action then we obtain a direct sum of copies of µ` .)(ii) Since J0 (p)[`](Qp ) and J0 (p)[`](Q) are of course equal, the Gal(Qp /Qp )action on HomT0m /` (W, J0 (p)[`](Qp )m ) = HomT0m /` (W, J0 (p)[`](Q)m ) extends to aGal(Q/Q)-action, and the semi-simplification of HomT0m /` (W, J0 (p)[`](Q)m ) as aGal(Q/Q)-module is a direct sum of copies of ρm . (This follows from the fact that
this Gal(Q/Q)-module is a successive extension of copies of J0 (p)[m](Q), whichitself is a successive extension of copies of ρm , as we observed above.)Since m is non-Eisenstein, we see by [23, prop. 2.2] that ρm is not finite atp. Taking into account (i) and (ii) above we see that lemma 4.5 implies thatthe dimension of Hom(Xm0 /` ⊗T0m /` W, µ` ) over Z/` is at most one-half that of
HomT0m /` (W, J0 (p)[`](Qp )m ). A consideration of (4.8) together with lemma 4.4 now
shows that each of Hom(Xm0 /` ⊗T0m /` W, µ` ) and HomT0m /` (W, Xm0 /`) must be of exactly one-half the dimension of HomT0m /` (W, J0 (p)[`](Qp )m ) over Z/`, and that (4.8)
is exact on the right. This completes the proof of the theorem. Corollary 4.9. If m is a maximal ideal of T of the first type, then the dimensionof J0 (p)[m] over T0 /m is equal to twice the dimension of Xm0 /m over T0 /m.
Proof. This follows by taking W to be T0 /m in the proof of theorem 4.6. 
Corollary 4.10. If m is a maximal ideal of the first type of T of residue characteristic `, then Xm0 /` is a faithful T0m /`-module.
Proof. We see by theorem 4.6 that it suffices to show that J0 (p)[`](Qp )m is a faithful
T0m /`-module. For this, it suffices to prove that J0 (p)[`](Qp ) is a faithful T0 /`module. Suppose that T ∈ T0 annihilates J0 (p)[`](Qp ). Then T is divisible by ` in
End(J0 (p)). Since T0 is the full ring of endomorphisms of J0 (p) [15, prop. II.9.5],
we see that in fact T is divisible by ` in T0 . This proves that J0 (p)[`](Qp ) is afaithful T0 /`-module, and so completes the proof of the corollary. As a final consideration for this section, we recall the following result.
Theorem 4.11. If m is a maximal ideal of T of the first type then T0m is Gorenstein
if and only if J0 [m] has rank two over T0 /m.
Proof. We see from [15, cor. II.15.2] that the only situation in which both conditions
occurring in the statement of the theorem don’t automatically hold are when m isordinary of residue characteristic two. Thus we need only prove the theorem inthis case. We may furthermore assume that p > 2 (since X0 (2) has genus zero),and so coprime to the residue characteristic of m. The theorem then follows from[16] and [7, prop. 12.8]. Indeed the latter reference shows that the étale part of thecontravariant m-adic Tate module of J0 (p) is free of rank one over Tm , and this isprecisely the input required to deduce the theorem from [16]. (We remark that thereader may also refer to the proof of [7, prop. 1.10], which uses identical reasoning
to the argument of [16].) 
5. Proofs of the main results.
In this section we prove the results stated in section 0.

30

MATTHEW EMERTON

Proof of theorem 0.3. We wish to show that the morphism θ : X ⊗T X −→ N
has image equal to M. Since its image lies in M, and M has index ∆ in N byproposition 1.1, it will suffice to show that the cokernel of θ has order ∆. Since θ
is obtained by composing the morphism (3.10) induced by the pairing of definition
3.4 with the isomorphism of part (i) of proposition 1.3, it suffices to show that thecokernel of (3.10) has order ∆.
For this, it will suffice to show that for each prime number `, the image of themorphism(5.1)

X ⊗Z Z` ⊗T⊗Z Z` X ⊗Z Z` −→ HomZ` (T ⊗Z Z` , Z` ),obtained by tensoring the morphism (3.10) through by Z` over Z, and which isinduced by the pairing (3.18), has cokernel of order equal to the maximal power of` dividing ∆. Thus for the remainder of the proof we fix `, and devote ourselves toproving this statement.
If we set A = Z` , B = T ⊗Z Z` and U = V = X ⊗Z Z` , then the running
hypotheses referred to in the statement of proposition 2.8 are satisfied. We willverify that for any maximal ideal of T ⊗Z Z` , either condition (i) or condition (ii) in
the statement of proposition 2.8 is satisfied. For this, given any any maximal ideal mof T of residue characteristic `, we must show either that the pairing (3.18) becomes
perfect when restricted to Xm × Xm and that Xm /` is a faithful Tm /`-module, orthat Xm is a free Tm -module of rank one.
∼
If m is of the first kind, then corollary 3.20 (ii) (and the isomorphism Xm0 −→Xm ) shows that the pairing (3.18) is perfect when restricted to Xm × Xm , while∼
corollary 4.10 shows that Xm0 /` −→ Xm /` is a faithful Tm /`-module.
∼
If m is of the second kind, then XmEis −→ Xm is isomorphic to Z` , and so is a free∼∼Tm −→ TEism −→ Z` -module of rank one, while if m is Eisenstein, then theorem 4.2
shows that Xm is a free Tm -module of rank one.
Thus the hypotheses of proposition 2.8 are satisfied, and we deduce that thecokernel of (5.1) is isomorphic to the cokernel of the morphism X ⊗Z Z` −→HomZ` (X ⊗Z Z` , Z` ) induced by the pairing (3.18). Lemma 3.16 (i) shows thatthe cokernel of this morphism has order equal to the maximal power of ` dividing∆. This completes the proof of theorem 0.3. 
Proof of theorem 0.4. We wish to prove that the morphism T −→ EndT (X ) is
an isomorphism. It suffices to prove this after tensoring this morphism throughby Z` , for each prime number `. Lemma 2.6 shows that it suffices to prove thatX /` is a faithful T/`-module, for each prime number `. Equivalently, it suffices toshow that Xm /` is a faithful Tm /`-module for each maximal ideal m of T of residue
characteristic `. This was observed in the course of proving theorem 0.3. (When mis of the second type or Eisenstein, note that Xm /` is even a free Tm /`-module ofrank one, and so in particular is faithful.) Proof of theorem 0.5. Let ` denote the residue characteristic of m. Then Tm is afree Z` -module of finite rank, and so is Gorenstein if and only HomZ` (Tm , Z` ) isa free Tm -module of rank one. Now proposition 1.3 (i) provides an isomorphism∼
HomZ` (Tm , Z` ) −→ Nm , and so we see that Tm is Gorenstein if and only if Nm is
free of rank one. Proposition 1.9 shows that Nm is free of rank one if and only if
Mm is free of rank one. Putting these results together yields the equivalence of (i)and (iii).

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 31

The equivalence of (i)0 and (iii)0 is proved similarly (and is even more direct,
since N 0 = M0 ), while the equivalence of (iii) and (iii)0 is simply a restatement oftheorem 1.10.
If m is of the first type, then the equivalence of (ii) and (ii)0 is tautological,
while the equivalence of (ii)0 and (iii)0 is given by corollary 4.9 and theorem 4.11.
∼∼
0
If m is of the second type, then Tm −→ TEism −→ Z` , while Tm = 0, and similarly
∼
Eis ∼
0
Xm −→ Xm −→ Z` , while Xm = 0. Thus in this case the conditions (ii), (ii)0 , (iii)and (iii)0 are all trivially satisfied. If m is Eisenstein then (ii) and (ii)0 both hold,by theorem 4.2, while (iii)0 holds, by [15, cor. II.16.3].
The results proved suffice to establish all the equivalences claimed in the statement of the theorem. The final claim of the theorem follows from theorem 0.3 andthe fact that a surjective morphism between Tm -modules which are both free ofrank one is necessarily an isomorphism. Proof of theorem 0.6. We begin by proving part (i). As usual, we let n denote thenumerator of (p − 1)/12, expressed in lowest terms. We first note that M0 /I is oforder n, by proposition 1.15. Thus in order to prove the theorem, we must showthat the morphism(5.2)

X 0 ⊗T0 X 0 −→ HomZ (T, Z),

induced by the restriction of the pairing of definition 3.4 to X 0 , has a cokernel
which is annihilated by I and is of order equal to n. (Note that the morphism
appearing in the statement of theorem 0.6 is obtained by composing (5.2) with theisomorphism of proposition 1.3 (ii).) For this it suffices to show, for each prime `,that the morphism
X 0 ⊗Z Z` ⊗T0 ⊗Z Z` X 0 ⊗Z Z` −→ HomZ` (T ⊗Z Z` , Z` )obtained by tensoring (5.2) through with Z` over Z, which is induced by the pairing(3.19), has cokernel annihilated by I and of order equal to the maximal power of `dividing n.
The proof will be similar to that of theorem 0.3. Namely, we fix `, and apply
proposition 2.8, taking A = Z` , B = T0 ⊗Z Z` , and U = V = X 0 ⊗Z Z` . Thatthe hypotheses of proposition 2.8 are satisfied follows from corollaries 4.10 and 3.20(ii), and theorem 4.2 (just as in the proof of theorem 0.3). The desired conclusion
then follows from lemma 3.16 (ii).
As for the proof of part (ii), it follows by similar observations to those made in the
proof of theorem 0.4. Indeed, to prove that T0 −→ EndT0 (X 0 ) is an isomorphism,
it suffices to prove that we obtain an isomorphism after tensoring this morphism
through by Z` , for each prime number `. Lemma 2.6 shows that it suffices to provethat X 0 /` is a faithful T0 /`-module, for each prime number `. Equivalently, it
suffices to show that Xm0 /` is a faithful T0m /`-module for each maximal ideal of
T0 of residue characteristic `. That this is the case was observed in the course ofverifying the hypotheses of proposition 2.8 in the proof of part (i). 
Proof of theorem 0.10. Both claims of theorem 0.10 follow from lemma 2.4, theproperties of the T-module X and the T0 -module X 0 shown in the proofs of theorems 0.3 and 0.6, and the isomorphisms of proposition 1.3, once one notes thatthe pairings X × HomZ (X , Z) −→ Z and X 0 × HomZ (X 0 , Z) −→ Z are perfect bydefinition. 32

MATTHEW EMERTON

6. Some remarks on the case when Tm is non-gorenstein
We begin this section by discussing a general commutative algebra framework
within which our results can be interpreted.
Definition 6.1. Let A be a reduced Noetherian complete one-dimensional local
ring, and let Ω denote a dualizing module for A.
(i) We say that an A-module U is a weak local theta-characteristic for A if it
is locally free of rank one when localized away from the maximal ideal of A, and if∼
there is an isomorphism of A-modules U −→ HomA (U, Ω).
(ii) We say that U is a strong local theta-characteristic for A if it is a weak local
theta-characteristic, and if the natural map A → EndA (U ) is an isomorphism.
If A is Gorenstein, then we can take Ω = A, and so A itself is a strong local
theta-characteristic. If A is arbitrary, and we let Ã denote the normalization of A,
then HomA (Ã, Ω) is a dualizing sheaf for the regular ring Ã, and so is isomorphic to
Ã. Since Ã is isomorphic to A when localized away from the maximal ideal of A, we
see that Ã is a weak local theta-characteristic for A. (This example was explained
to the author by Kovács.) However, since Ã acts on itself through multiplication
via A-linear endomorphisms, we see that Ã is not a strong local theta-characteristic
for A unless A = Ã; that is, unless A itself is normal.
If m is any maximal ideal of the first type in T, then the perfect pairing ofcorollary 3.20 (ii) and the adjointness isomorphism (2.2) give rise to an isomorphism∼
Xm0 −→ HomT0m (Xm0 , HomZ` (T0m , Z` )). Since HomZ` (T0m , Z` ) is a dualizing module
for the one-dimensional local ring T0m , this isomorphism together with theorem 0.4
shows that Xm0 is a strong local theta-characteristic for T0m ; in particular, the ringT0m does admit a strong local theta-characteristic. The preceding discussion showsthat this is a weaker commutative algebra condition then being Gorenstein. In
order to understand how much weaker, it is natural to ask the following question:
does any reduced Noetherian complete one-dimensional local ring admit a strong
local theta-characteristic? I don’t know the answer.
We next apply the results of the preceding sections to draw some conclusions
about the structure of the m-adic Tate module of J0 (p) when Tm is non-Gorenstein.
Let us suppose that m is an ordinary maximal ideal of T of the first type, ofresidue characteristic ` 6= p. Then the proof of [7, 12.10] and the discussion of [16]show that the m-adic Tate module Tm J0 (p) of J0 (p) (that is, the localization at m ofthe `-adic Tate module of J0 (p)) sits in a short exact sequence of T0m [Gal(Q` /Q` )]modules(6.2)

0 −→ Tm J0 (p)0 −→ Tm J0 (p) −→ Tm J0 (p)e −→ 0,

induced by the connected-étale short exact sequence of the `-divisible group of theNéron model of J0 (p) over Z` . Furthermore, Tm J0 (p)0 is isomorphic to T0m as a
T0m -module, while Tm J0 (p)e is isomorphic to HomZ` (T0m , Z` ).
Proposition 6.3. The short exact sequence (6.2) splits as a short exact sequenceof T0m -modules if and only if T0m is Gorenstein.
Proof. If T0m is Gorenstein, then HomZ` (T0m , Z` ) is isomorphic to T0m , and so (6.2)
is an extension of free T0m -modules, and hence splits. Thus to complete the proof ofthe proposition, it remains to show that (6.2) is non-split if T0m is not Gorenstein.

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 33

Consider the short exact sequence(6.4)

0 −→ J0 (p)[`]0m −→ J0 (p)[`]m −→ J0 (p)[`]em −→ 0

obtained by tensoring (6.2) through by Z/` over Z` . We will show that (6.4) is
non-split as a short exact sequence of T0m /`-modules.
Recall from theorem 4.6 that J0 (p)[`]m is isomorphic to Xm0 /` ⊕ Xm0 /` as a T0m /`module. Thus (6.4) is isomorphic (as a short exact sequence of T0m /`-modules) toa short exact sequence of the form
0 −→ T0m /` −→ Xm0 /` ⊕ Xm0 /` −→ HomZ/` (T0m /`, Z/`) −→ 0.
It follows from the following lemma that there are no injective morphisms of T0m /`modules from HomZ/` (T0m /`, Z/`) to Xm0 /` ⊕ Xm0 /`, and so such a short exact sequence is necessarily non-split. 
Before proving the next lemma, we note that if m is a maximal ideal of T of
residue characteristic `, then the surjection T0m /` −→ T0m /m allows us to regard
HomZ/` (T0m /m, Z/`) as a submodule of HomZ/` (T0m /`, Z/`).
Lemma 6.5. If m is a maximal ideal of T of residue characteristic ` for which T0m
is not Gorenstein, then any morphism from HomZ/` (T0m /`, Z/`) to Xm0 /` contains
HomZ/` (T0 /m, Z/`) in its kernel.
Proof. Note that since T0m is assumed to be non-Gorenstein, theorem 1.14 showsthat the maximal ideal m is necessarily of the first type. We will write A = Z/`,
B = T0m /`, and U = Xm0 /`. Then A is a field, B is a non-Gorenstein Artinian
local A-algebra, and U is a B-module which is of the same dimension as B as an
A-vector space, and which is isomorphic to its A-dual HomA (U, A) as a B-module.
(This follows from corollary 3.20 (ii), since m is of the first type.)
Let n denote the maximal ideal of B. We will show that if A, B and U satisfy theconditions laid out in the preceding paragraph, then any morphism of B-modules
φ : HomA (B, A) −→ U necessarily contains HomA (B/n, A) in its kernel. Since
HomA (B/n, A) is a one-dimensional B/n vector space, if it is not in the kernel of φthen φ induces an injection HomA (B/n, A) −→ U. Dualizing φ over A, we obtain amorphism ψ : HomA (U, A) −→ B which induces a surjection HomA (U, A) −→ B/n.
Thus ψ is itself a surjection, and since its source and target are of the same dimen∼sion over A, ψ is an isomorphism. But this is impossible, since HomA (U, A) −→ U
is an A-self-dual B-module, while B is not, being non-Gorenstein by assumption.
We conclude that HomA (B/n, A) is in the kernel of φ, as claimed. 
Proposition 6.3 should be contrasted with the fact that Tm J0 (p) sits in a short
exact sequence of T0m [Gal(Qp /Qp )]-modules
0 −→ HomZ` (Xm0 , Z` (1)) −→ Tm J0 (p) −→ Xm0 −→ 0,
which is split as a short exact sequence of T0m -modules (as is proved by an obvious extension of the argument that proves theorem 4.6). (Of course, the proof of
proposition 6.3 makes it clear that it is precisely because this latter exact sequence
splits that the exact sequence (6.2) does not split when T0m is non-Gorenstein.)
We close this section by describing concretely the T0m /2-module Xm0 /2 in theparticular case when m is a non-Gorenstein maximal ideal of residue characteristic

34

MATTHEW EMERTON

2 for which T0m /2 has length four. Writing k = T0m /m, we note that T0m /2 is thenisomorphic to k [x, y, z]/(x2 , y 2 , z 2 , xy, xz, yz). (One checks that this is the unique
non-Gorenstein local ring of length four of characteristic 2 having residue field k).
When p = 431 or 503, T has a maximal ideal m for which the completion T0m is
non-Gorenstein and of rank four over Z2 [14], and so our description applies inparticular to these cases.
Lemma 6.6. If T0m is non-Gorenstein and T0m /2 is of length four, then Xm0 /2 [m]has rank two over Z/2, and is equal to mXm0 /2.
Proof. Write d for the dimension of Xm0 /2 [m] over k. Applying lemma 4.4 with
W = T0m /m shows that Xm0 /m also has dimension d. The short exact sequence
0 −→ mXm0 /2 −→ Xm0 /2 −→ Xm0 /m −→ 0then shows that mXm0 /2 has dimension 4 − d. Since m2 T0m /2 = 0, we see that
mXm0 /2 ⊂ Xm0 /2 [m], and so Xm0 /2 [m] has dimension at least 4 − d. Thus we seethat d ≥ 2.
Since corollary 4.10 shows that Xm0 /2 is a faithful T0m /2-module, we see that thenatural map mT0m /2 → EndT0m /2 (Xm0 /2) is injective. The image of this map lies in
HomT0m /2 ((Xm0 /2)/(Xm0 /2 [m]), mXm0 /2).
Since mT0m /2 has dimension three, while HomT0m /2 ((Xm0 /2)/(Xm0 /2 [m]), mXm0 /2) has
dimension (4 − d)2 , we see that d ≤ 2. Combining this calculation with that of thepreceding paragraph, we see that d = 4 − d = 2, and so deduce the lemma. 
The T0m /2 action on Xm0 /2 is determined by the action of the maximal ideal
mT0m /2 on Xm0 /2. This action in turn is determined by the composite(6.7) mT0m /2 −→ HomT0m /2 ((Xm0 /2)/(Xm0 /2 [m]), mXm0 /2)∼= Homk (Xm0 /m, Xm0 /2 [m]) −→ Xm0 /2 [m] ⊗k Xm0 /2 [m].
The final isomorphism holds because lemma 4.4 shows that the pairing Xm0 /2 ×
Xm0 /2 −→ Z/2 given by corollary 3.20 (ii) induces a duality between the k-vector
spaces Xm0 /2 [m] and Xm0 /m. Since the pairing of corollary 3.20 (ii) is symmetric, themap (6.7) has image lying in the subspace Sym2k (Xm0 /2 [m]) of symmetric tensors.
It is also injective (as was observed in the proof of lemma 6.6) and so is an isomorphism, since mT0m /2 and Sym2k (Xm0 /2 [m]) both have dimension three over T0 /m.
Thus the T0m /2-module structure on Xm0 /2 is entirely determined by choosing an∼
isomorphism of three-dimensional k-vector spaces mT0m /2 −→ Sym2T0 /m (Xm0 /2 [m]).
This gives a complete description of the structure of Xm0 /2 as a T0m /2-module inthe case when T0m /2 is non-Gorenstein of length four.
7. The case of non-prime level
In this section, we will discuss the extension of our results to the case of non-primelevel. Unfortunately, the lack of control over Eisenstein primes in this situation, as
well as the possibility of primes of fusion between old and new forms, means that
our results are weaker than in the case of prime level. This is the primary reason
for relegating them to this final section of the paper.

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 35

We begin by generalizing the notation introduced in sections 1 and 3. If N
is a positive integer we write N (N ) for the space of weight two modular forms∞Xf =an q n on Γ0 (N ) for which an ∈ Z (n ≥ 1), N (N )0 for the submodule ofn=0

N (N ) consisting of cusp forms, and N (N )Eis for the submodule of N (N ) spannedby Eisenstein series.
We let T denote the finite Z-algebra of Hecke operators acting on N (N ), T0
denote the quotient of T that acts faithfully on N (N )0 , and TEis denote the quotientof T that acts faithfully on N (N )Eis .
We have the following generalization of proposition 1.3 (whose proof can befound in the same reference given for the proof of that proposition).
∼

Proposition 7.1. (i) There is a natural isomorphism of T-modules N (N ) −→HomZ (T, Z), defined by the pairing hf, T i = a1 (f |T ).
∼(ii) There is a natural isomorphism of T0 -modules N (N )0 −→ HomZ (T0 , Z),
defined by the same pairing as that of part (i).
Now suppose that N = M p, where p is a prime which does not divide M . We
let Vp denote the “degeneracy operator” f (q) 7→ f (q p ), which yields a morphism
N (M ) −→ N (M p). The Z-submodule N (M ) + N (M )|Vp of N (N ) is stable underT, and we let Told denote the quotient of T that acts faithfully on it. We let
N (N )old denote the maximal T-submodule of N (N ) on which the T-action factors
through Told . Then N (N )old is the saturation of N (M ) + N (M )|Vp in N (N ), andso it contains N (M ) + N (M )|Vp with finite index. Similarly, we write N (N )0old =Eis
N (N )0 ∩ N (N )old and N (N )Eis
∩ N (N )old . (Here the subscript “old”
old = N (N )stands for “old at p”.)
The quotient of N (N )Eis by N (N )Eis
old is a free Z-module of rank one. Let E
denote the weight two Eisenstein series on Γ0 (p), let Σ = {`1 , `2 , . . . , `t } denote theset of primes dividing M , and set
XX
Enew = E 0 (q) −E 0 (q ` ) +E 0 (q `α `β ) − · · · + (−1)t E 0 (q `1 ···`t ).
`∈Σ`α ,`β ∈Σ1≤α<β≤t(Here and below, the subscript “new” stands for “new at p”.) Then Enew is anormalized T-eigenform characterized by the conditions (1 + `)Enew if ` does not divide N ,Enew |T` = `Enew if ` divides M ,Enew if ` = p,
as ` ranges over all primes. One easily sees that Enew does not lie in N (N )Eis
old , andso its image generates the quotient N (N )Eis /N (N )Eis
.
old
Eis
We let TEis
which acts faithfully on Enew . As a ring,new denote the quotient of T0
TEis
new is isomorphic to Z. We let N (N )new denote the maximal Z-submodule of
N (N )0 orthogonal to N (N )0old under the Petersson inner product, and write T0new
for the quotient of T0 which acts faithfully on N (N )0new . We also write Tnew for0
the image of T in TEisnew ⊕ Tnew , and write N (N )new for the maximal T-submoduleof N (N ) on which the T-action factors through Tnew (so that N (N )new is thesaturation in N (N ) of the submodule spanned by N (N )0new and Enew ). We have

36

MATTHEW EMERTON

defined N (N )new in such a way that the natural map N (N )new ⊕N (N )old → N (N )is an injection of T-modules whose cokernel is finite.
The modular curve X0 (N ) (regarded as a scheme over Zp ) has semi-stable reduction modulo p. Its geometric fibre in characteristic p consists of two irreducible
components. The specializations of the cusps 0 and ∞ of X0 (N ) lie on different
components; let us label the two components C0 and C∞ , according to which ofthese two specializations they contain. Each of these components is naturally isomorphic to X0 (M )/Fp (which is a smooth curve, because p does not divide M ).
The two components meet transversally at the points corresponding to supersingular elliptic curves. More precisely, if the point x on C∞ corresponds to the pair
(E, C) consisting of a supersingular elliptic curve E and a cyclic subgroup of E oforder M , then x is identified with the point on C0 obtained by taking the Frobenius
twist of the pair (E, C). Let S = {x0 , . . . , xs } denote the collection of singularpoints on X0 (N )/Fp . By regarding each xi as a point on C∞ , we may regard it ascorresponding to a pair (Ei , Ci ) as above, well determined up to isomorphism.
We let X denote the Z-module of divisors supported on the set S, and let X 0
denote the submodule of X consisting of divisors of degree zero. There is a naturalaction of T on X and X 0 , which makes X a faithful Tnew -module, and makes X 0 afaithful T0new -module.
Let J0 (N ) denote the Jacobian of X0 (N )/Q . It is an abelian variety over Q, thespecial fibre of whose Néron model over Zp has a semi-abelian connected component(that is, is the extension of an abelian variety by a torus). We let T0 (N ) denotethe maximal torus of this semi-abelian variety. There is a natural identification of
X 0 with the character lattice of T0 (N ).
There is a monodromy pairing h , i : X × X −→ Z, defined by the formula1
hxi , xj i = δi,j # Aut((Ei , Ci )). This morphism is not T-bilinear if M > 1. How2
ever, it is easily modified to be so. As usual, let wM denote the Atkin-Lehner involution on X0 (M ), and define [ , ] : X × X −→ Z by the formula [x, y] = hx, wM yi.
The morphism [ , ] is T-bilinear, and so induces a morphism of T-modules(7.2)

X −→ Hom(X , Z),

which is injective (and hence has finite cokernel). (For the assertions of the last
several paragraphs concerning the geometry of X0 (N ), we refer to sections 2 and 3of [20], and the references cited therein.)
The discussion of section 2 (in particular the adjointness isomorphism (2.2))
together with proposition 7.1 shows that (7.2) induces a T-linear morphism∼θ : X ⊗T X −→ HomZ (T, Z) −→ N (N ).
In fact, since X is a Tnew -module, the image of θ lies in N (N )new . Similarly to thecase when M = 1, this morphism is closely related to the construction of certain
theta series from quaternion algebras. However, the details are more complicatedwhen M > 1.
If xi and xj are two elements of S, corresponding to the pairs (Ei , Ci ) and(Ej , Cj ) of supersingular elliptic curves and cyclic subgroups of order M , thenwe define Li,j = Hom((Ei , Ci ), (Ej , Cj )) to be the Z-submodule of Hom(Ei , Ej )consisting of those homomorphisms φ : Ei −→ Ej for which φ(Ci ) ⊂ Cj , equipped

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 37

with the positive definite quadratic form given by taking the degree of an isogeny.
The quadratic space Li,j is rank four of level N = M p. (In other words, if L∗i,jdenotes the dual lattice to Li,j , the cokernel of the injection Li,j → L∗i,j inducedby the quadratic form on Li,j has exponent M p.)
If d is a divisor of M, then for any cyclic subgroup C of order M , we let C[d]denote the unique subgroup of C of order d. Then, given xi and xj as in the(d)preceding paragraph, define the quadratic space Li,j to be(d)Li,j = Hom((Ei /Ci [d], Ci /Ci [d]), (Ej , Cj [M/d])),regarded as the quadratic subspace of Li,j consisting of homomorphisms containing
Ci [d] in their kernel.
As above, let Σ = {`1 , · · · , `t } be the collection of prime factors of M .
Proposition 7.3. If xi and xj are any two elements of S, thenθ(xi ⊗ wM xj ) =

X1
1
(`)
Θ(Li,j ) −Θ(Li,j )
2
2
`∈Q

+

1
1
(` ` )(` ···` )Θ(Li,jα β ) − · · · + (−1)t Θ(Li,j1 t ).
22`α ,` ∈QX

β
1≤α<β≤t

Proof. If n is a positive integer, thenan (θ(xi ⊗ wM xj )) = a1 (θ(xi ⊗ wM xj )|Tn ) = a1 (θ(Tn xi ⊗ wM xj ) = [Tn xi , wM xj ]1
= hTn xi , xj i = the number of cyclic n-isogenies φ : Ei −→ Ej for which2ker(φ) ∩ Ci is trivial and φ(Ci ) equals Cj
1= the number of φ ∈ Li,j for which ker(φ) ∩ Ci is trivial.
2
If φ is an element of Li,j for which ker(φ) ∩ Ci 6= 0, then ker(φ) ⊃ Ci [q] for some(q)
prime q dividing M , and so φ ∈ Li,j . The formula of the proposition now follows
directly by inclusion-exclusion counting. We remark that, analogously to propositions 3.11 and 3.15, this result gives analgebro-geometric proof that certain linear combinations of theta series are modular
forms. In fact, if M > 1 then the modular forms in N (N )new have vanishingconstant term, and so in contrast to the situation of proposition 3.15, the questionof finding an algebro-geometric determination of the constant term of a theta-series
does not arise.
It is natural to ask for a description of the image of θ. Proposition 7.3 showsthat answering this question is closely related to describing the Z-submodule of
N (N ) spanned by theta series. We begin by observing that it is easy to answer thequestion after tensoring through by Q over Z.
Proposition 7.4. The morphism
X ⊗Z Q ⊗T⊗Z Q X ⊗Z Q −→ N (N )new ⊗Z Q

38

MATTHEW EMERTON

induced by θ is an isomorphism of free Tnew ⊗Z Q-modules of rank one.
Similarly, the morphism
X 0 ⊗Z Q ⊗T0 ⊗Z Q X 0 ⊗Z Q −→ N (N )0new ⊗Z Q
induced by θ is an isomorphism of free T0new ⊗Z Q-modules of rank one.
Proof. The ring Tnew ⊗Z Q is a reduced Q-algebra (as follows from the results of[1]), and both X ⊗Z Q and N (N )new ⊗Z Q are faithful Tnew ⊗Z Q-modules of thesame dimension over Q as Tnew ⊗Z Q. Thus they are free Tnew ⊗Z Q-modules ofrank one. Since the pairing X ⊗Z Q × X ⊗Z Q −→ Q induced by [ , ] is perfect,
lemma 2.4 yields the first claim of the proposition. The second can be proved in asimilar fashion, or by localizing the isomorphism just proved at the direct summandT0new ⊗Z Q of Tnew ⊗Z Q. 
This result has the following corollary, originally due to Hijikata and Saito [12].
Corollary 7.5. The Q-vector space N (N ) ⊗Z Q is spanned by N (N )old ⊗Z Q to(d)gether with the theta series Θ(Li,j ), as xi and xj range over the basis S of X , andd ranges over all products of primes from the set Q.
Proof. This follows from propositions 7.3 and 7.4 and the fact that the naturalmorphism N (N )old ⊕ N (N )new −→ N (N ) becomes an isomorphism after tensoring
through with Q over Z. 
As for describing the image of θ in the Z-module N (N ), when M > 1 we are
not as successful as in the case when M = 1. Just as in that case, we would liketo proceed by analyzing the morphism induced by θ after completing at all the
maximal ideals m of T. If m is such a maximal ideal of residue characteristic `,then there is a semi-simple Galois representation ρm : Gal(Q/Q) −→ GL2 (T/m)
attached in a canonical fashion to m, characterized by the fact that for any primeq not dividing N `, the representation ρm is unramified at q, and the characteristic
polynomial of ρm (Frobq ) is equal to X 2 − Tq X + q. Unfortunately we do not havecontrol over the completion Xm without making some restrictive hypotheses onboth m and ρm , and these same hypotheses are then required to draw inferences
concerning the image of the map induced by θ.
Theorem 7.6. Let m be a maximal ideal of T which is p-new and for which ρm isirreducible. Suppose in addition that(i) the residue characteristic ` of m is odd, and `2 does not divide N ;(ii) m is not a prime of fusion between `-old and `-new forms in N (N )0 (this isautomatic if ` does not divide N , since then N (N )0 contains no `-new forms);(iii) if m is a prime of fusion between p-old and p-new forms in N (N )0 , then atleast one of the following holds: (a) ` does not divide p − 1, or (b) ρm (Frobp ) does
not act as a scalar.
Then the morphism Xm ⊗Tm Xm −→ (N (N )new )m induced by θ is an isomorphismof free (Tnew )m -modules of rank one.
Proof. We first note that since ρm is irreducible, the natural maps Tm −→ T0m ,(Tnew )m −→ (T0new )m , and Xm0 −→ Xm are isomorphisms. We next observe thatXm ⊗Z Q is free of rank one over (Tnew )m ⊗Z Q, by proposition 7.4. Suppose we
can prove that X /m is one-dimensional over T/m = Tnew /m; then since (Tnew )mSUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 39

is reduced (by the results of [1]) we conclude that Xm is a free (Tnew )m -module ofrank one. From [20, thm. 3.12/prop. 3.14/thm. 3.22] we deduce that the pairing
Xm ×Xm −→ Z` induced by [ , ] is perfect. Lemma 2.4 now shows that the morphism
Xm ⊗Tm Xm −→ (N (N )new )m is surjective. Since it becomes an isomorphism after
tensoring with Q over Z, by proposition 7.4, and since its source is a free (Tnew )m module of rank one, it must be an isomorphism between its source and target. This
proves the theorem.
It remains to be shown that X /m is one-dimensional over T/m. If ` is prime to N ,this follows from [21, prop. 1] and hypothesis (iii) in the statement of the theorem.
(As already noted, when ` is prime to N , hypothesis (ii) is superfluous.) Now
suppose that ` exactly divides N . Hypothesis (ii) and the discussion of [22, p. 4]
shows that ρm appears with multiplicity one in J0 (N )[m](Q). If ρm is not p-old,then the main theorem of [20] (which applies to our situation even though ` divides
N , since we have the requisite statement regarding multiplicity one) shows that ρm
is not finite at p, and the same argument as used to prove [23, thm. 2.3] showsthat X /m is one-dimensional over T/m. If, on the other hand, ρm is p-old, then by
assumption (ii) ` 6= p, and the proof of [21, prop. 1] then extends to again showthat X /m is one-dimensional over T/m (the point as usual being that we have therequisite multiplicity one statement for J0 (N )[m](Q)). 
Let S denote the following set of primes:
S = {` | `2 divides 4N }∪ {` | ` is a congruence prime between N (N )Eis and N (N )0 }∪ {` | ` is a congruence prime between `-old and `-new forms in N (N )0 }∪ {` | ` is a congruence prime between p-old and p-new forms in N (N )0 }.
Corollary 7.7. The Z-module spanned by N (N )0old , together with the image of themap X 0 ⊗T0 X 0 −→ N (N )0new induced by θ, has index in N (N )0 divisible only byprimes in S.
Proof. The set S contains all primes which occur as the residue characteristics ofmaximal ideals m of Tnew which do not satisfy the hypotheses of theorem 7.6, and sothat theorem shows that the cokernel of the morphism X 0 ⊗T0 X 0 −→ N (N )0new hasorder divisible only by primes in S. Furthermore, the index of N (N )0new + N (N )0old
in N (N )0 is divisible precisely by the primes of congruence between N (N )0new
and N (N )0old , and these are also contained in S. The theorem follows from theseobservations. Condition (ii) of theorem 7.6, and hence the conclusion of the previous corollary,can be improved slightly by appealing to [22, thm. 2]. For example, suppose thatp = 11 and M = 3, so that N = 33. In this case there are no 11-old cusp forms, andso N (33)0new = N (33)0 and T0new = T0 . The set S consists of the primes two, five(these are the congruence primes between N (N )Eis and N (N )0 ) and three (which
is a congruence prime between the 3-old and 3-new forms in N (33)0 ). However,since the 3-new quotient of J0 (33) is an elliptic curve, [22, cor. to thm. 2] showsthat any maximal ideal of residue characteristic three satisfies multiplicity one, andso the proof of theorem 7.6 extends to cover these maximal ideals. We conclude
that the set S in corollary 7.7 can be replaced by the smaller set S 0 = {2, 5}. An
explicit calculation shows in fact that the image of the map X 0 ⊗T0 X 0 −→ N (33)0
is of index ten in N (33)0 .

40

MATTHEW EMERTON

Appendix. Supersingular points and p-adic
periods of weight two modular forms.
In this appendix we continue with the notation of section 7. In particular, we
work in level N = M p, with p not dividing M . We also assume that if M = 1 thenp ≥ 5 (so that p is then prime to 2∆, where as before ∆ denotes the denominatorof (p − 1)/12). Let Ω(cusps) denote the dualizing sheaf of X0 (N ) over Zp , twistedby the invertible sheaf corresponding to the relative divisor of cusps in X0 (N )over Zp . The Hecke correspondences act naturally on H 0 (X0 (N )/Zp , Ω(cusps)) (see[15, §II.6] for a discussion in the case when M = 1) and this action serves tomake H 0 (X0 (N )/Zp , Ω(cusps)) a T ⊗Z Zp -module. In fact, one has the more precise
statement given by the following result.
Proposition A.1. The Zp -module H 0 (X0 (N )/Zp , Ω(cusps)) is naturally a T⊗Z Zp submodule of N (N ) ⊗Z Zp containing N (N )new ⊗Z Zp .
Proof. We refer to [7, §8] for a discussion of the dualizing sheaf Ω of X1 (N ) overZp [ζ] (where ζ is a primitive pth root of unity) and its relation to weight-two cusp
forms on Γ1 (N ). (Note that our N is Gross’s N p.) In particular, [7, prop 8.4] showsthat H 0 (X1 (N )/Zp [ζ] , Ω) is a Zp [ζ]-lattice inside the space of weight two cusp formson Γ1 (N ) defined over Qp (ζ), which consists precisely of those cusp forms f suchthat both f and f |wζ have q-expansion coefficients lying in Zp [ζ].
An identical argument to the one that proves this result applies in our situation toshow that H 0 (X0 (N )/Zp , Ω(cusps)) is a Zp -lattice inside N (N ) ⊗Zp Qp , consisting
precisely of those modular forms for which both f and f |wp have q-expansion
coefficients lying in Zp . (Here as usual wp denotes the Atkin-Lehner involution at p
on N (N )⊗Z Qp = N (M p)⊗Z Qp .) Consequently, H 0 (X0 (N ), Ω(cusps)) is containedin N (N ), and the action of the Hecke correspondences on H 0 (X0 (N )/Zp , Ω(cusps))
is obtained by restricting the T ⊗Z Zp -action on N (N ) ⊗Z Zp . We thus conclude
that H 0 (X0 (N )/Zp , Ω(cusps)) is a T ⊗Z Zp -submodule of N (N ) ⊗Z Zp .
To show that H 0 (X0 (N )/Zp , Ω(cusps)) contains N (N )new ⊗Z Zp , it suffices toshow that if f ∈ N (N )new ⊗Z Zp , then both f and f |wp have q-expansion coefficients
lying in Zp . By definition of N (N ), all q-expansion coefficients of f lie in Zp , except
perhaps the zeroth; that is to say, the constant term of f . However this constantterm vanishes if M > 1, while if M = 1 then we have assumed that p does notdivide 2∆, and so this constant term lies in Zp in this case as well, by virtue of
proposition 1.2. To see that all q-expansion coefficients of f |wp also lie in Zp , wenote that f |wp = −f |Tp if f ∈ N (N )new ⊗Z Zp (by an obvious modification of[1, lem. 7] or [20, prop. 3.7], each of which proves this in the cuspidal case). 
Recall that S = {x0 , . . . , xs } denotes the set of singular points on the special
fibre of X0 (N ), that X denotes the free Z-module spanned by S, and that eachpoint xi ∈ S corresponds to a supersingular elliptic curve Ei over Fp together with1
a cyclic subgroup Ci of Ei of order M . We write ei = # Aut((Ei , Ci )).
2
Let W denote the Witt ring of Fp . After base-changing from Z to W , one finds(see the discussion of [20, p. 439]) that the formal completion of X0 (N ) at the point
xi in its special fibre is isomorphic to the formal spectrum ofW [[x, y]]/(xy − pei ) = W [[x,

pei
]].
x

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 41

Denote this ring by Ai . By writing the ring Ai in terms of x rather than y, we are
implicitly making a choice of orientation of the annulus obtained after passing to
the rigid analytic fibre over K = W ⊗Z Q of Spf Ai (this annulus is the tube of thepoint xi in the rigid analytic space underlying X0 (N )/K ). As explained in [2, §7],
there is a canonical choice of orientation of this annulus, so that the “outer edge”
is closer to the cusp ∞, while the “inner edge” is closer to the cusp 0. We may anddo assume that the variables are labelled so that the differential dx/x yields thiscanonical orientation. The global sections of the dualizing sheaf of Spf Ai are a freep ei
W [[x,]]-module of rank one spanned by dx/x, which we denote Ω(Ai ), and wexhave a map resi : Ω(Ai ) −→ W obtained by taking the residue of a section.
Proposition A.2. The morphism(A.3)

H 0 (X0 (N )/W , Ω(cusps)) −→ X ⊗Z W,

given by the formula ω 7→
modulesnewi resi (ω| Spf Ai )xi , induces an isomorphism of TP∼

H 0 (X0 (N )/W , Ω(cusps))/(H 0 (X0 (N )/W , Ω(cusps)) ∩ N (N )old ⊗Z W ) −→ X ⊗Z W.
Proof. That the map (A.3) is a morphism of T-modules follows from the compatibility of the action of pulling-back or pushing-forward a differential via a map with thetaking of residues of this differential. The elements of H 0 (X0 (N )/W , Ω(cusps)) ∩N (N )old ⊗Z W are rational multiples of pull-backs of differentials from X0 (M ), a
curve with good reduction modulo p, and so have trivial residues along the supersingular annuli of X0 (N ). Thus (A.3) does induce a map
H 0 (X0 (N )/W , Ω(cusps))/(H 0 (X0 (N )/W , Ω(cusps)) ∩ N (N )old ⊗Z W ) −→ X ⊗Z W.
The T-actions on both the source and target of this map factor through Tnew , andthe source and target are both free W -modules of equal (finite) rank. Thus to showthat this map is an isomorphism of Tnew -modules, it suffices to prove that the map
obtained by reducing (A.3) modulo p is surjective.
As remarked by Gross in [7, §8], the arguments of [15, §II.3] prove that thenatural map H 0 (X0 (N )/W , Ω(cusps)) ⊗W k −→ H 0 (X0 (N )/k , Ω(cusps)) is an isomorphism. Let C∞ and C0 denote the components of X0 (N )/k containing thecusps ∞ and 0 respectively; they are both isomorphic to X0 (M )/k . A section ω ofH 0 (X0 (N )/k , Ω(cusps)) is determined by giving a pair of differentials of the thirdkind (ω∞ , ω0 ), one on each of C∞ and C0 , regular away from the points xi and thecusps, and such that the residues of each differential at each of the points xi are
negative to one another.
The reduction modulo p of the map (A.3) is given by(ω∞ , ω0 ) 7→

X

resxi (ω∞ )xi ∈ X ⊗Z k.

i

Since the set of cusps of both C∞ and C0 are non-empty, the residue theorem showsthat this map is surjective, as required. 

42

MATTHEW EMERTON

Note that when M = 1, there are no p-old forms in N (N ) = N (p), and so∼
proposition A.2 yields an isomorphism N (p) ⊗Z W −→ X ⊗Z W. The argumentused to prove this result makes it clear that it provides a kind of p-adic lift of themod p situation considered by Ohta in [18, §2]. This result also provides a new
proof of proposition 3.1, and (following [18]) can be used to give an alternative
proof that M ⊗Z Zp is spanned by the theta series Θ(Li,j ). (A result stronger thanEichler’s theorem, but weaker than our theorem 0.3.)
Continuing to consider the case M = 1, note that if we consider the particularelement E ∈ N (p) ⊗Z W, then lemma 3.13 shows that the image of E under theisomorphism of proposition A.2 must be equal to wx, for some element w ∈ W. Let
C denote the open subset of the irreducible component C∞ of X0 (p)/k obtained byremoving the cusp ∞ together with the supersingular points, and let X denote theopen affinoid contained in X0 (p)/K which is the preimage of C under specialization.
Thus X is isomorphic to the complement of the supersingular residue class disks inthe affine j-line over K. If we restrict E to X we may regard it as a differential on
X. The residue theorem shows that the sum of its residues around the puncturesof X is equal to zero, and so we find that a0 (E) + w deg(x) = 0. Proposition 3.15
implies that w = −1/2∆, and hence that the residue of E around the tube ofxi is equal to −1/2ei . This result (or to be precise, its analogue when X0 (p) is
replaced by the curve that classifies p-isogenies of elliptic curves equipped withfull level-two structure) was first obtained by Stevens [25, prop. 1]. Both Steven’s
argument and ours rely on the residue theorem together with a knowledge of theconstant term of the Eisenstein series. As observed in the remarks following theproof of proposition 3.15, it would be interesting to have a purely algebro-geometricdetermination of the constant w that appears above.
Proposition A.2 is closely related to Coleman’s p-adic Shimura isomorphism.
Indeed, if we tensor the isomorphism of proposition A.2 with the fraction field K
of W and restrict to cusp forms on the left-hand side, and hence to X 0 on the∼right-hand side, we obtain an isomorphism N (N )0new ⊗Z K −→ X 0 ⊗Z K, which
is precisely the isomorphism obtained by taking k = 2 in [2, thm. 9.1] (up to theharmless replacement of Coleman’s Γ1 (M ) ∩ Γ0 (p) level-structure by our Γ0 (M p)level structure). In some sense this is a p-adic analogue of the Riemann relations:
it shows that the p-new differentials on X0 (N )/K are uniquely determined by theirperiods along the supersingular annuli.
We note that proposition A.2 yields a variant of the main theorem of [17].
Corollary A.4. If m is a maximal ideal of residue characteristic p in T for which(Told )m = 0, and either p ≥ 5 or the Galois representation ρm is irreducible, then∼there is a natural isomorphism of Tm -modules N (N )m −→ Xm . Consequently bothare free Tm -modules of rank one, and so Tm is Gorenstein.
Proof. Since (Told )m = 0 by assumption, each of the morphisms Tm −→ (Tnew )m ,(N (N )new )m −→ N (N )m , andH 0 (X0 (N )/W , Ω(cusps))m −→H 0 (X0 (N ), Ω(cusps))m /(H 0 (X0 (N )/W , Ω(cusps))m ∩ (N (N )old )m ⊗Zp W )are isomorphisms. Thus propositions A.1 and A.2, combined with the flat base∼change isomorphism H 0 (X0 (N )/Zp , Ω(cusps)) ⊗Zp W −→ H 0 (X0 (N )/W , Ω(cusps)),SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 43

yield an isomorphism of Tm -modules
(A.5)

∼

(N (N )new )m ⊗Zp W −→ Xm ⊗Zp W.

Now the isomorphism of proposition 7.1 induces an isomorphism∼(A.6) (N (N )new )m ⊗Zp W −→ HomZp ((Tnew )m , W )∼−→ HomW ((Tnew )m ⊗Zp W, W ).
Furthermore, the morphism (7.2) induces an isomorphism∼

Xm −→ HomZp (Xm , Zp ).

(A.7)

Indeed, if p ≥ 5 then this follows from the fact that the cokernel of the map (7.2)
has exponent dividing twelve, while if ρm is irreducible, we observed in the proof oftheorem 7.6 that it is implied by [20, thm. 3.12/prop. 3.14/thm. 3.22].
Tensoring the isomorphism (A.7) by W over Zp , and combining it with theisomorphisms (A.5) and (A.6), we find that Tm ⊗Zp W is a Gorenstein ring, andhence that the same is true of Tm . 
We make one final remark in this appendix, again restricting ourselves to thecase when M = 1. If we combine proposition A.2 with theorem 0.5 (noting, as has
just been proved, that T ⊗Z W is Gorenstein) we obtain an isomorphism of freeT ⊗Z W -modules of rank one
(A.8)

∼

(N (p) ⊗Z W ) ⊗T⊗Z W (N (p) ⊗Z W ) −→ N (p) ⊗Z W,∼

or equivalently, an isomorphism T⊗Z W −→ N (p)⊗Z W. Thus we see that N (p)⊗Z W
is a free T ⊗Z W -module of rank one, equipped with a canonical generator, which
we denote by P .
The isomorphism (A.8) determines (and is determined by) a non-degenerate Tsymmetric W -valued pairing on N (p) ⊗Z W . Explicitly,hω, νi =X

ei resxi (ω| Spf Ai ) resxi (ν| Spf Ai ).

i

It is interesting to compare this pairing with that of [3], which is defined precisely
on those forms on Γ1 (p) whose traces to Γ0 (p) vanish.
The generator P is characterized in terms of this pairing by the conditionhω, P i = a1 (ω) for any element ω of N (p) ⊗Z W . Thus P could be thought ofas a p-adic analogue of a Poincaré series.
References
1. Atkin, A. O. L., Lehner, J., Hecke operators on Γ0 (m), Math. Ann. 185 (1970), 134–160.
2. Coleman, R. F., A p-adic Shimura isomorphism and p-adic periods of modular forms, Contemp. Math. 165 (1994), 21–51.
3. Coleman, R. F., A p-adic inner product on elliptic modular forms, Barsotti symposium inalgebraic geometry, Academic Press, 1994, pp. 125–151.
4. Eichler, M., Über die Idealklassenzahl total definiter Quaternionenalgebren, Math. Z. 43(1938), 102–109.

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MATTHEW EMERTON

5. Eichler, M., Über die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew.
Math. 195 (1955), 156–171.
6. Gross, B. H., Heights and the special values of L-series, Number theory (Montreal, Que.,
1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc., 1987, pp. 115–187.
7. Gross, B. H., On a tameness criterion for Galois representations associated to modular forms(mod p), Duke Math. J. 61 (1990), 445–517.
8. Gross, B. H., Course at Harvard University, Spring, 1996.
9. Grothendieck, A., SGA VII, Exposé IX, SLN, vol. 288, 1972, pp. 313–523.
10. Hartshorne, R., Residues and Duality, SLN 20 (1966).
11. Hecke, E., Lectures on the theory of algebraic numbers, Springer-Verlag, 1981.
12. Hijikata, H., Saito, H., On the representability of modular forms by theta series, Number
theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya,
Tokyo, 1973, pp. 13–21.
13. Igusa, J. -I., Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad.
Sci. USA 44 (1958), 312–314.
14. Kilford, L., Some examples of non-Gorenstein Hecke algebras associated to modular forms,
preprint, available at http://www.ma.ic.ac.uk/~ljpk/maths/maths.html.
15. Mazur, B., Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Etud. Sci. 47(1977), 33–186.
16. Mazur, B., letter to K. Ribet and J. Tilouine, in “Hecke algebras and the Gorenstein property”, by J. Tilouine, Modular forms and Fermat’s last theorem (G. Cornell, J. H. Silverman,
G. Stevens, eds.), Springer-Verlag, 1997.
17. Mazur, B., Ribet, K. A., Two-dimensional representations in the arithmetic of modular curves,
Astérisque 196–197 (1991), 215–255.
18. Ohta, M., Theta series mod p, J. Fac. Sci. Tokyo 28 (1981), 679–686.
19. Ribet, K. A., Mod p Hecke operators and congruences between modular forms, Invent. Math.
71 (1983), 193–205.
20. Ribet, K. A., On modular representations of Gal(Q/Q) arising from modular forms, Invent. Math. 100 (1990), 431–476.
21. Ribet, K. A., Multiplicities of Galois representations in Jacobians of Shimura curves, Israel
Math. Conf. Proc. 3 (1990), 221–236.
22. Ribet, K. A., Multiplicites of p-finite mod p Galois representations in J0 (N p), Bol. Soc. Brasil.
Mat. (N. S.) 21 (1991), 177–188.
23. Ribet, K. A., Torsion points on J0 (N ) and Galois representations, Arithmetic Theory of
Elliptic Curves (J. Coates, ed.), SLN, vol. 1716, 1999.
24. Serre, J. -P., Sur les représentations modularies de degré 2 de Gal(Q/Q), Duke Math. J. 54(1987), 179–230.
25. Stevens, G., Coleman’s L-invariant and families of modular forms, preprint (1996).
Northwestern University
Department of Mathematics
2033 Sheridan Rd.
Evanston, IL 60208-2730, USA
E-mail address: emerton@math.northwestern.edu