SUPERSINGULAR ELLIPTIC CURVES, THETA

SERIES AND WEIGHT TWO MODULAR FORMS

Matthew Emerton

Northwestern University

Abstract. Let p be a prime, and let M denote the space of weight two modular forms on Γ_{0}(p) all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that M is spanned as a ℤ-module by theta series attached to the unique quaternion algebra that is ramified at p, at infinity, and at no other primes.

This paper deals with two subjects and their interaction. The first is the problem of 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, and remark that the isomorphism class of L as a quadratic space depends only on the left ideal class of L in its maximal order.) Eichler [5] proved that the theta series of L is a weight two modular form on Γ_{0}(p), and that as L ranges over a collection of 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 Γ_{0}(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 other coefficients 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 of left 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 Γ_{0}(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 X_{0}(p) in characteristic p. The key properties of X, which imply the above result on theta series, are that the natural map 𝕋 ⟶ End_{𝕋}(X) is an isomorphism,

Typeset by AMS-TEX

2000 Mathematics Subject Classification: 11F11, 11F27, 11F37

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 the analogous statements after tensoring with ℚ, for they then follow from the fact that 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 ⟶ End_{𝕋⊗_{ℤ}ℚ}(X ⊗_{ℤ} ℚ) is an isomorphism.

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 observed by 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 the structure 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 of modular 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 and 4 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 an interpretation 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 the case 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 X_{0}(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 in nature. 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, by appealing instead to level lowering results. We hope that both ideas may have additional 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 like to thank Robert Coleman for interesting conversations on related topics during the Summer 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 the anonymous referee for their helpful comments on various preliminary versions of this paper, which helped to improve both the mathematics and the exposition.

SUPERSINGULAR ELLIPTIC CURVES AND WEIGHT TWO MODULAR FORMS 3

0. Statement of results.

We will describe our results in more detail. We begin by introducing some notation and filling in some background. If f is a weight two modular form on Γ_{0}(p),

we let a_{n}(f) denote the n^{th} q-expansion coefficient of f, so that f = ∑_{n=0}^{∞} a_{n}(f)q^{n}.

We let M denote the ℤ-module of weight two modular forms f on Γ_{0}(p) for which

a_{n}(f) ∈ ℤ when n ≥ 1 and a_{0}(f) ∈ \frac{1}{2}ℤ.

One way of producing elements in M is as follows: if E_{i} and E_{j} are two supersingular elliptic curves in characteristic p, then L_{i,j} = Hom(E_{i}, E_{j}) is a free ℤ-module of rank four equipped with a positive definite quadratic form, given by taking the degree of an isogeny (see proposition 3.6 below for details). If Θ(L_{i,j}) denotes the

theta series of L_{i,j}, then \frac{1}{2}Θ(L_{i,j}) lies in M. (See proposition 3.15 below, as well as the subsequent remarks, for a discussion of this point.) The quadratic spaces L_{i,j} are precisely those attached to the quaternion algebra B in the manner described in the introduction (as is explained in [6, §2]).

Let X_{0}(p) denote the curve over Spec ℤ that is a coarse moduli space for the Γ_{0}(p)-moduli problem. The fibres of X_{0}(p) are smooth curves over every point of Spec ℤ other than p. By contrast, if g denotes the (constant) arithmetic genus of the fibres of X_{0}(p) (which we recall is approximately equal to p/12, as follows from proposition 3.15 below, for example), then the geometric fibre of X_{0}(p) in characteristic p is the union of two rational curves meeting at g+1 ordinary double points. Let x_{0}, …, x_{g} denote these singular points; they are in bijective correspondence with the isomorphism classes of supersingular curves E_{i}. Let X denote the free ℤ-module of divisors supported on the x_{i}, and define a ℤ-bilinear pairing

(0.1)  X × X ⟶ M

by the formula x_{i} × x_{j} ↦ \frac{1}{2}Θ(L_{i,j}).

As above, we let 𝕋 denote the ℤ-algebra of Hecke operators acting on the space of weight two modular forms on Γ_{0}(p). The ℤ-module M is closed under the action of 𝕋, and so M is naturally a faithful 𝕋-module. The action of the Hecke correspondences on the set of x_{i} 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 ⟶ M

from the tensor product of X with itself over 𝕋 to M.

Theorem 0.3. The morphism (0.2) is surjective. Equivalently, M is spanned by

the half-theta series \frac{1}{2}Θ(L_{i,j}).

Recall that X and M are both free of rank g + 1 over ℤ (where g is equal to the genus of X_{0}(p)), and so in particular (taking into account the symmetry L_{i,j} ⟶^{∼} L_{j,i}; see lemma 3.7) there are (g + 1)(g + 2)/2 theta series Θ(L_{i,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 the following result.

Theorem 0.4. The natural morphism 𝕋 ⟶ End_{𝕋}(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 (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 e_{i} denote one-half the number of automorphisms of E_{i}, then this pairing is defined as follows: ⟨x_{i}, x_{j}⟩ = e_{i}δ_{i,j}. The connection with (0.2) is provided by

the formula ⟨x_{i}, x_{j}⟩ = a_{1}(\frac{1}{2}Θ(L_{i,j})). (See section 3 for a more complete discussion of 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 M^{0} denote the 𝕋-submodule of M consisting of those f for which a_{0}(f) = 0 (that is, the submodule of cusp forms in M), and let 𝕋^{0} denote that quotient of 𝕋 that acts faithfully on M^{0}. We let X^{0} denote the submodule of X consisting of divisors of degree zero. Then X^{0} is also a faithful 𝕋^{0}-module (see theorem 3.1 below).

For any maximal ideal 𝔪 of 𝕋 and any 𝕋-module U we let U_{𝔪} denote the completion of U at 𝔪. In particular, 𝕋_{𝔪} denotes the completion of 𝕋 at 𝔪, and U_{𝔪} is naturally a 𝕋_{𝔪}-module.

Theorem 0.5. Fix a maximal ideal 𝔪 of 𝕋. The following are equivalent: (i) The 𝕋_{𝔪}-module M_{𝔪} is free of rank one. (i)^{0} The 𝕋_{𝔪}^{0}-module M_{𝔪}^{0} is free of rank one. (ii) The 𝕋_{𝔪}-module X_{𝔪} is free of rank one. (ii)^{0} The 𝕋_{𝔪}^{0}-module X_{𝔪}^{0} is free of rank one. (iii) The ring 𝕋_{𝔪} is Gorenstein. (iii)^{0} The ring 𝕋_{𝔪}^{0} is Gorenstein. If these equivalent conditions hold, then the morphism X_{𝔪} ⊗_{𝕋_{𝔪}} X_{𝔪} ⟶ M_{𝔪}

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 𝕋_{𝔪} and 𝕋_{𝔪}^{0} 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 described in 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 Eisenstein case. In particular, in this case theorem 0.5 should properly be regarded, both in its 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 examples of non-Eisenstein 𝔪 for which 𝕋_{𝔪} is non-Gorenstein. (The results of [15] show that such 𝔪 are necessarily of residue characteristic two and ordinary.) Thus in this case theorem 0.5 is genuinely a list of equivalent possibilities, that may or may not hold in any particular case. (It is mentioned in [14] that William Stein has implemented theorem 0.5 as a means of testing whether or not 𝕋_{𝔪} is Gorenstein, by taking advantage of the fact that condition (ii) is amenable to being checked by computer algebra.)

Since the modules X and X^{0} appear with equal status among the conditions of theorem 0.5, one might ask whether suitable analogues of theorems 0.3 and 0.4 hold when X is replaced by X^{0} and M by M^{0}. We will state a result that answers this question, after making a definition. Let 𝕋^{Eis} denote the quotient of 𝕋 which acts faithfully on the weight two Eisenstein series on Γ_{0}(p) (so that after forgetting the 𝕋-algebra structure, 𝕋^{Eis} is isomorphic to ℤ). Let I denote the ideal in 𝕋 which is the kernel of the surjection 𝕋 ⟶ 𝕋^{Eis}. Then the ideal I𝕋^{0} of 𝕋^{0} has finite index in 𝕋^{0}, equal to the numerator of (p − 1)/12 (when written in lowest terms) (see proposition 1.8, part (iii)); denote this number by n. Note that this ideal I𝕋^{0} is the celebrated Eisenstein ideal of [15].

Theorem 0.6. (i) The image of the morphism X^{0} ⊗_{𝕋^{0}} X^{0} ⟶ M^{0} induced by (0.2) is equal to IM^{0}, and is of index n in M^{0}.

(ii) The natural map 𝕋^{0} ⟶ End_{𝕋^{0}}(X^{0}) is an isomorphism.

As a final variant on our theme, we note that the ℤ-valued pairing on X described above yields an embedding X ⟶ Hom_{ℤ}(X , ℤ), whose image has finite index in Hom_{ℤ}(X , ℤ). Thus we may extend the pairing (0.1) to a pairing

(0.7)  X × Hom_{ℤ}(X , ℤ) ⟶ M ⊗_{ℤ} ℚ.

If x̌_{0}, …, x̌_{g} denotes the basis of Hom_{ℤ}(X , ℤ) dual to the basis x_{0}, …, x_{g} of X, then x̌_{j} = \frac{x_{j}}{e_{j}} for j = 0, …, g, and so the pairing (0.7) maps the pair (x_{i}, x̌_{j}) to

the normalized theta series \frac{Θ(L_{i,j})}{2e_{j}}. Thus, if we let N denote the submodule of

M ⊗_{ℤ} ℚ consisting of modular forms f for which a_{n}(f) ∈ ℤ if n ≥ 1 (but with no condition on a_{0}(f)), then (0.7) takes values in N, and induces a morphism

(0.8)  X ⊗_{𝕋} Hom_{ℤ}(X , ℤ) ⟶ N .

There is an analogue for X^{0}, yielding a morphism

(0.9)  X^{0} ⊗_{𝕋^{0}} Hom_{ℤ}(X^{0}, ℤ) ⟶ M^{0}.

(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

spanned over ℤ by the normalized theta series \frac{Θ(L_{i,j})}{2e_{j}}.

We owe the idea of studying the 𝕋-module X and the map (0.2) entirely to Gross, and the preceding results answer (in a slightly refined form) questions raised by 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 from propositions 1.1 and 1.3 that there is a natural injection M ⟶ Hom_{ℤ}(𝕋, ℤ) whose cokernel has order equal to the denominator of (p − 1)/12, which hence becomes an isomorphism after inverting 6. Theorem 0.5 thus shows that after inverting 6 the morphism (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] ⟶^{∼} Hom_{ℤ[1/6]}(𝕋[1/6], ℤ[1/6]).

In particular, Hom_{ℤ[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 in a finite extension of ℚ, then the inverse different D^{−1} of O (which by definition is isomorphic to Hom_{ℤ}(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 Hom_{ℤ}(𝕋, ℤ) is a square element in the the Picard group of 𝕋; or whether Hom_{ℤ[1/2]}(𝕋[1/2], ℤ[1/2]) is a square 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 the elements of Pic(𝕋) corresponding to M, N, X, and Hom_{ℤ}(X , ℤ), and describe the maps (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 Γ_{0}(p)

is two-dimensional, spanned by the Eisenstein series E = \frac{5}{12} + q + 3q^{2} + ⋯ and

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-module

spanned by \frac{6(E − f)}{5} and f, while N is the free ℤ-module spanned by \frac{E − f}{5} and f.

We now provide an explicit description of the Hecke ring 𝕋. Each of 𝕋^{Eis} and 𝕋^{0} is isomorphic to ℤ, and (since E and f are congruent modulo five) the image of 𝕋 under the injection 𝕋 ⟶ 𝕋^{Eis} ⊕ 𝕋^{0} is equal to {(m_{1}, m_{2})|m_{1} ≡ m_{2} (mod 5)}. This ring is Gorenstein; indeed, the pairing

(m_{1}, m_{2}) × (m′_{1}, m′_{2}) ↦ \frac{m_{1}m′_{1} − m_{2}m′_{2}}{5}

yields an isomorphism of 𝕋 with its ℤ-dual. (Since f is supersingular at two, this is a special case of theorem 1.14 below.) One computes that Pic(𝕋) is isomorphic to the cokernel of the natural map ℤ^{×} ⟶ (ℤ/5)^{×}, which has order two. The module U = {(m_{1}, m_{2})|2m_{1} ≡ m_{2} (mod 5)} (with the obvious 𝕋-action) represents the non-trivial element of Pic(𝕋).

Define a map of 𝕋-modules 𝕋 ⟶ M via (m_{1}, m_{2}) ↦ m_{1} \frac{6E}{5} − m_{2} \frac{f}{5} . This 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

(m_{1}, m_{2}) ↦ m_{1} \frac{E}{5} − m_{2} \frac{f}{5} is an isomorphism of 𝕋 with N, showing that N also represents the trivial element of Pic(𝕋). (This last statement also follows from part (i) of proposition 1.3 below, together with the preceding observation that 𝕋 is isomorphic to its ℤ-dual.) With respect to these two isomorphisms, the inclusion M ⊂ N corresponds to the endomorphism of 𝕋 defined via (m_{1}, m_{2}) ↦ (6m_{1}, m_{2}).

The two supersingular j-invariants in characteristic 11 are j = 0 and j = 1728. We denote the corresponding basis elements of X by x_{0} and x_{1}. Then e_{0} = 3 and e_{1} = 2. Write x = 2x_{0} + 3x_{1}. It follows from the results of section 3 below that 𝕋 acts on x through 𝕋^{Eis} and on x_{1} − x_{0} through 𝕋^{0}. Define a map of 𝕋-modules U ⟶ X via (m_{1}, m_{2}) ↦ m_{1} \frac{x}{5} + m_{2} \frac{x_{1} − x_{0}}{5} . It is immediate that this map is an isomorphism, and so X represents the non-trivial element of Pic(𝕋).

Let x̌_{0} and x̌_{1} denote the basis of Hom_{ℤ}(X , ℤ) which is dual to the basis x_{0}, x_{1} of X. Then the degree morphism deg : X ⟶ ℤ is equal to the element x̌_{0} + x̌_{1} of Hom_{ℤ}(X , ℤ), while the element 2x̌_{1}−3x̌_{0} annihilates x. It follows from the results of section 3 below that 𝕋 acts on deg through 𝕋^{Eis} and on 2x̌_{1} −3x̌_{0} through 𝕋^{0}. Thus

the morphism U ⟶ Hom_{ℤ}(X , ℤ) defined via (m_{1}, m_{2}) ↦ m_{1} \frac{deg}{5} + m_{2} \frac{2x̌_{1} − 3x̌_{0}}{5} is an isomorphism of 𝕋-modules, and Hom_{ℤ}(X , ℤ) also represents the non-trivial element of Pic(𝕋).

With respect to the isomorphisms U ⟶^{∼} X and U ⟶^{∼} Hom_{ℤ}(X , ℤ) constructed in the preceding two paragraphs, the tautological perfect pairing X × Hom_{ℤ}(X , ℤ) ⟶ ℤ corresponds to the perfect pairing U × U ⟶ ℤ defined by

(m_{1}, m_{2}) × (m′_{1}, m′_{2}) ↦ \frac{m_{1}m′_{1} + m_{2}m′_{2}}{5} .

The embedding X ⟶ Hom_{ℤ}(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 (m_{1}, m_{2}) ↦ (6m_{1}, m_{2}).

Recall that L_{i,j} denotes the quadratic space of endomorphisms from the elliptic curve E_{i} representing x_{i} to the elliptic curve E_{j} representing x_{j}. From the values of e_{0} and e_{1} we can determine that

\frac{1}{2}Θ(L_{0,0}) = \frac{1}{2} + 3q + ⋯ = \frac{6E + 9f}{5} ,

that

\frac{1}{2}Θ(L_{0,1}) = \frac{1}{2} + 0q + ⋯ = \frac{6(E − f)}{5},

and that

\frac{1}{2}Θ(L_{1,1}) = \frac{1}{2} + 2q + ⋯ = \frac{6E + 4f}{5} .

From these formulas we compute the following particular values of the map X ⊗_{𝕋} X ⟶ M given by (0.2):

x ⊗ x ↦ 30E,
x ⊗ (x_{1} − x_{0}) ↦ 0,

and

(x_{1} − x_{0}) ⊗ (x_{1} − x_{0}) ↦ 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)  (m_{1}, m_{2}) ⊗ (m′_{1}, m′_{2}) ↦ (m_{1}m′_{1}, −m_{2}m′_{2}),

which witnesses the fact that U yields an element of Pic(𝕋) of order two. We also see that the induced map X^{0} ⊗_{𝕋^{0}} X^{0} ⟶ M^{0} = 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} Hom_{ℤ}(X , ℤ) ⟶ N given by (0.8):

x ⊗ deg ↦ 5E,
x ⊗ (2x̌_{1} − 3x̌_{0}) ↦ 0,
(x_{1} − x_{0}) × deg ↦ 0,

and

(x_{1} − x_{0}) × (2x̌_{1} − 3x̌_{0}) ↦ 5f.

From these values we deduce that, in terms of the given isomorphisms U ⟶^{∼} X , U ⟶^{∼} Hom_{ℤ}(X , ℤ), and 𝕋 ⟶^{∼} N , the map (0.8) also correspond to the isomorphism U ⊗_{𝕋} U ⟶^{∼} 𝕋 given by (0.11). If we note that Hom_{ℤ}(X^{0}, ℤ) is the

free ℤ-module of rank one spanned by \frac{2x̌_{1} − 3x̌_{0}}{5} , we also deduce that the map

X^{0} ⊗_{𝕋^{0}} Hom_{ℤ}(X^{0}, ℤ) ⟶ M^{0} 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

on Γ_{0}(p) for which a_{n}(f) lies in ℤ (n ≥ 1) and a_{0}(f) lies in \frac{1}{2}ℤ, and let 𝕋 denote the ℤ-algebra of Hecke operators, which (tautologically from its construction) acts faithfully on M. Also, M^{0} denotes the submodule of cusp forms in M (that is, those f for which a_{0}(f) = 0), and 𝕋^{0} the quotient of 𝕋 that acts faithfully on M^{0}.

The tensor product M ⊗_{ℤ} ℚ is naturally identified with the space of weight two modular forms on Γ_{0}(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 a_{n}(f) lies in ℤ when n ≥ 1, and for notational consistency we will let N^{0} denote M^{0} when regarded as the submodule of cusp forms in N.

Let E = \frac{p − 1}{24} + ∑_{n=1}^{∞} σ′(n)q^{n} (where σ′(n) = ∑_{{d|n \atop (p,d)=1}} d) denote the weight two

Eisenstein series on Γ_{0}(p). Then E is a 𝕋-eigenform, and so defines a surjection 𝕋 ⟶ ℤ that sends the Hecke operator T_{n} to the integer σ′(n). We let 𝕋^{Eis} denote ℤ regarded as a quotient of 𝕋 in this way. We denote by N^{Eis} the 𝕋-submodule of N spanned by the Eisenstein series E, and by M^{Eis} the intersection M ∩ N^{Eis}. These are both free modules of rank one over 𝕋^{Eis}, and M^{Eis} (respectively N^{Eis}) is the maximal submodule of M (respectively N) on which the 𝕋-action factors through 𝕋^{Eis}.

We define the positive integers n and Δ to be respectively the numerator and denominator of the fraction (p−1)/12, when reduced to lowest terms. Since a_{0}(E) = (p − 1)/24, we see immediately that ΔE lies in M, and generates M^{Eis}.

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 result shows that this index is exactly equal to Δ.

Proposition 1.1. Let A_{0} : N ⟶ ℚ denote the map f ↦ 2Δa_{0}(f). Then A_{0} 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)  [figure: 0; 0; 0; 0; M^{0}; N^{0}; 0; 0; 0; M; N; A_{0}; M/N; ∼; 0; 0; Δℤ; ℤ; ℤ/Δ; 0; 0; 0; 0; .]

In this diagram, the action of 𝕋 on the objects of the bottom row factors through 𝕋^{Eis}.

Proof. If p = 2 or 3, then N = N^{Eis} is spanned by E, and M = M^{Eis} is spanned by ΔE. 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 A_{0} 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 Γ_{0}(p). Note that in this reference B(ℤ) denotes the submodule of N consisting of modular forms all of whose q-expansion coefficients lie in ℤ, while B^{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 a_{0}(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 lies in 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/Δ), we see that this is equivalent to b dividing 2Δa; since a and b are coprime, we deduce that b divides 2Δ. Thus A_{0}(f) = 2Δa/b ∈ ℤ.

The preceding paragraph shows that A_{0} is a map N ⟶ ℤ. Immediately from its definition, we see that N^{0} is the kernel of A_{0}. In order to complete the proof that (1.2) has exact rows and columns we have to show that A_{0} yields a surjection of N onto ℤ, and restricts to yield a surjection of M onto Δℤ.

Note that the Eisenstein series E maps to the integer

A_{0}(E) = 2Δa_{0}(E) = 2Δ\frac{p − 1}{24} = n.

On the other hand we can find g ∈ M such that a_{0}(g) = 1/2 (via the theta series construction of section 3, for example), and so

A_{0}(g) = 2Δa_{0}(g) = Δ.

Thus we see that the image of M does equal Δℤ, while the image of N contains g.c.d(n, Δ) = 1, and so equals ℤ.

Finally, we have to show that the action of 𝕋 on the bottom row factors through 𝕋_{Eis}. 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)] shows that if p ≥ 5 and m is an integer prime to p, then 1 is the q-expansion of a modular form for Γ_{0}(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 shows that 1 is the q-expansion of an element of B(ℤ/m) if and only if m divides 2Δ.

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 ⟶^{∼} Hom_{ℤ}(𝕋, ℤ),

defined by the pairing ⟨f, T ⟩ = a_{1}(f|T).

(ii) There is a natural isomorphism of 𝕋^{0}-modules

M^{0} = N^{0} ⟶^{∼} Hom_{ℤ}(𝕋^{0}, 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 of Γ_{0}(p). A simple modification of the proof of this theorem yields (i). The key point is 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 ⟶ 𝕋^{Eis} ⊕ 𝕋^{0}.

This is ℤ-dual to the injection

(1.5)  0 ⟶ N^{Eis} ⊕ N^{0} ⟶ N .

We also have the short exact sequence

(1.6)  0 ⟶ N^{0} ⟶ N ⟶^{A_{0}} ℤ ⟶ 0

which makes up the central column of (1.2).

The map A_{0} : N ⟶ ℤ corresponds by proposition 1.3 to an element of 𝕋, which we denote by t_{0}, characterized by the property a_{1}(f|t_{0}) = A_{0}(f) = 2Δa_{0}(f) for any f ∈ N. Dualizing (1.6) yields the short exact sequence

(1.7)  0 ⟶ ℤt_{0} ⟶ 𝕋 ⟶ 𝕋^{0} ⟶ 0.

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

Proposition 1.8. (i) The kernel of the surjection 𝕋 ⟶ 𝕋^{0} is principal, generated by t_{0}.

(ii) The element t_{0} − n of 𝕋 lies in the kernel of the surjection 𝕋 ⟶ 𝕋^{Eis}.

(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 the fact that A_{0}(E) = 2Δa_{0}(E) = n, while a_{1}(E) = 1. To prove claim (iii), it suffices to prove the same statement for the cokernel of the injection (1.5). A consideration of the short exact sequence (1.6) shows that the cokernel of (1.5) is isomorphic to A_{0}(N)/A_{0}(N^{Eis}) = ℤ/A_{0}(E) = ℤ/n. This proves the proposition. □

The idea of using the surjectivity of A_{0} 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 ⟶ 𝕋^{Eis} ⊕ 𝕋^{0} is not an isomorphism unless n = 1 (in which case p = 2, 3, 5, 7 or 13, and so 𝕋^{0} = 0); this is the main theme of [15]. We regard Spec 𝕋 as being the union of its closed subschemes Spec 𝕋^{Eis} and Spec 𝕋^{0}. From this optic, the maximal ideals 𝔪 of 𝕋 are seen to be of three types. The first type consists of those that lie in Spec 𝕋 \ Spec 𝕋^{Eis}; that is, they are the maximal ideals that induce the unit ideal of 𝕋^{Eis}, but a non-unit ideal of 𝕋^{0}. The second type consists of those that lie in Spec 𝕋 \ Spec 𝕋^{0}; that is, they are the maximal ideals that induce the unit ideal of 𝕋^{0}, but a non-unit ideal of 𝕋^{Eis}. The third type consists of those that lie in the intersection of Spec 𝕋^{Eis} and Spec 𝕋^{0}; that is, they are the maximal ideals that induce a non-unit ideal of both 𝕋^{Eis} and 𝕋^{0} (the primes of fusion between 𝕋^{Eis} and 𝕋^{0} 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 characteristics of the Eisenstein primes are precisely the prime numbers that divide n. (This calculation was first carried out in [15, p. 96].) This will be quite important in the arguments to come, as we see already in the next proposition.

Proposition 1.9. For any maximal ideal 𝔪 of 𝕋, the completion M_{𝔪} is free of rank one as a 𝕋_{𝔪}-module if and only the completion N_{𝔪} is free of rank one as a 𝕋_{𝔪}-module.

Proof. Let ℓ denote the residue characteristic of 𝔪. If 𝔪 is of the first type (that is, generates the unit ideal of 𝕋^{Eis}), then completing (1.2) at 𝔪 yields an isomorphism M_{𝔪} ⟶^{∼} N_{𝔪}, and the corollary is clear in this case. If 𝔪 is of the second type (that is, generates the unit ideal of 𝕋^{0}), then completing (1.2) at 𝔪 yields isomorphisms M_{𝔪} ⟶^{∼} Δℤ_{ℓ} and N_{𝔪} ⟶^{∼} ℤ_{ℓ}. Thus both are free 𝕋_{𝔪} ⟶^{∼} 𝕋_{𝔪}^{Eis} ⟶^{∼} ℤ_{ℓ}-modules of rank one, so that the corollary is also clear in this case. If 𝔪 is an Eisenstein ideal, then ℓ divides n, and so does not divide Δ (since n and Δ are coprime by construction). Thus completing (1.2) at 𝔪 yields an isomorphism M_{𝔪} ⟶^{∼} N_{𝔪}, 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 𝕋_{𝔪} is Gorenstein if and only if 𝕋_{𝔪}^{0} is Gorenstein.

Proof. If 𝔪 is a maximal ideal of the first type, then 𝕋_{𝔪} ⟶^{∼} 𝕋_{𝔪}^{0}, and so the theorem is true in this case. If 𝔪 is a maximal ideal of the second type then 𝕋_{𝔪} ⟶^{∼} 𝕋_{𝔪}^{Eis}, and so 𝕋_{𝔪}^{0} = 0 (and hence is Gorenstein). On the other hand, if ℓ denotes the residue characteristic of 𝔪 then 𝕋_{𝔪}^{Eis} ⟶^{∼} ℤ_{ℓ}. Since this ring is also Gorenstein, the theorem follows in this case.

It remains to treat the case when 𝔪 is Eisenstein. In this case, let I denote the kernel of the map 𝕋_{𝔪} ⟶ 𝕋_{𝔪}^{Eis} and J denote the kernel of the map 𝕋_{𝔪} ⟶ 𝕋_{𝔪}^{0} ⟶^{∼} ℤ_{ℓ}. Then I ∩ J = 0, and so Spec 𝕋_{𝔪} is the scheme-theoretic union of Spec 𝕋_{𝔪}^{Eis} and Spec 𝕋_{𝔪}^{0}. The scheme-theoretic intersection of 𝕋_{𝔪}^{Eis} and 𝕋_{𝔪}^{0} is certainly a Cartier divisor on Spec 𝕋_{𝔪}^{Eis} ⟶^{∼} Spec ℤ_{ℓ}, while [15, thm. II.18.10] implies that it is a Cartier divisor on Spec 𝕋_{𝔪}^{0}.

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 𝕋_{𝔪} is Gorenstein if and only if Spec 𝕋_{𝔪}^{Eis} and Spec 𝕋_{𝔪}^{0} are Gorenstein. Since Spec 𝕋^{Eis} ⟶^{∼} Spec ℤ_{ℓ} is Gorenstein, 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 a dualizing complex F^{•}. Suppose that X_{1} and X_{2} are two closed subschemes of X such that X is the scheme-theoretic union of X_{1} and X_{2}, and such that the schemetheoretic intersection X_{1} ∩ X_{2} is a Cartier divisor on both X_{1} and X_{2}. Then X is Gorenstein if and only if each of X_{1} and X_{2} are Gorenstein.

Proof. Recall that a finite-dimensional connected Noetherian scheme Y with dualizing complex F_{Y}^{•} is Gorenstein if and only if F_{Y}^{•} is quasi-isomorphic to a locally free O_{Y} -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]), and dim(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, they must 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 the neighbourhood of points in the complement of X_{1} ∩ X_{2} (since in such a neighbourhood X is simply the disjoint union of X_{1} and X_{2}), we may in addition assume that x lies in X_{1} ∩ X_{2}. By shrinking the neighbourhood under consideration, we may and do assume that X_{1}, X_{2} and X_{1} ∩ X_{2} are connected.

Write D = X_{1} ∩ X_{2}. Let i_{1} : X_{1} ⟶ X and i_{2} : X_{2} ⟶ X denote the closed immersions of X_{1} and X_{2} respectively into X. Let I_{1} and I_{2} denote the ideal sheaves of O_{X} cutting out X_{1} and X_{2} respectively. Then I_{1} ∩ I_{2} = 0. (This is the meaning of X being the scheme-theoretic union of X_{1} and X_{2}.) In particular, the natural map I_{1} ⟶ (I_{1} +I_{2})/I_{2} is an isomorphism. The target of this isomorphism is the ideal of the Cartier divisor D on X_{2}, and so is a locally free O_{X_{2}}-module of rank one. Thus I_{1} is a locally principal ideal sheaf in O_{X}, with annihilator equal to I_{2}. Similarly, I_{2} is a locally principal ideal sheaf in O_{X}, with annihilator equal to I_{1}. By restricting our attention to a smaller neighbourhood of x, if necessary, we may and do assume that both I_{1} and I_{2} are in fact principal, generated by sections a_{1} and a_{2} of O_{X} respectively. Thus we have a short exact sequence

(1.12)  0 ⟶ i_{1∗}O_{X_{1}} ⟶^{·a_{2}} O_{X} ⟶ i_{2∗}O_{X_{2}} ⟶ 0,

and an analogous sequence obtained by reversing the roles of X_{1} and X_{2}.

If we write F_{1}^{•} = i_{1}^{!}F^{•} and F_{2}^{•} = i_{2}^{!}F^{•}, then we obtain dualizing complexes on X_{1} and X_{2} respectively [10, prop. V.2.4]. Applying RHom(–, F) to (1.12) (and remembering that RHom(j_{∗}O_{Y} , F^{•}) = j^{!}F^{•} for any closed immersion j : Y ⟶ X [10, thm. III.8.7(3)]) yields the distinguished triangle

(1.13)  ⋯ ⟶ F_{2}^{•} ⟶ F^{•} ⟶^{·a_{2}} F_{1}^{•} ⟶ ⋯

in the derived category of coherent O_{X}-modules. Again, we obtain an analogous distinguished triangle by reversing the roles of X_{1} and X_{2}.

Let us suppose that X_{1} and X_{2} are Gorenstein. The remark made at the beginning of the proof implies that X_{1} and X_{2} are of the same dimension, say n (and so X is also of dimension n). Thus F_{1}^{•} and F_{2}^{•} are quasi-isomorphic to rank one locally free sheaves over O_{X_{1}} and O_{X_{2}} 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 O_{X}-module supported in degree −n, and obtain the short exact sequence

0 ⟶ H^{−n}(X_{2}, F_{2}^{•}) ⟶ H^{−n}(X, F^{•}) ⟶^{·a_{2}} H^{−n}(X_{1}, F_{1}^{•}) ⟶ 0.

Reversing the roles of X_{1} and X_{2} we similarly obtain a short exact sequence

0 ⟶ H^{−n}(X_{1}, F_{1}^{•}) ⟶ H^{−n}(X, F^{•}) ⟶^{·a_{1}} H^{−n}(X_{2}, F_{2}^{•}) ⟶ 0.

Combining these two exact sequence, we find that the injection H^{−n}(X_{1}, F_{1}^{•}) ⟶ H^{−n}(X, F^{•}) identifies H^{−n}(X_{1}, F_{1}^{•}) with the annihilator of a_{1} in H^{−n}(X, F^{•}), and also that multiplication by a_{2} induces a surjection of H^{−n}(X, F^{•}) onto this annihilator. Again, similar remarks apply after reversing the roles of X_{1} and X_{2}.

If we write M = H^{−n}(X, F^{•}), then we may summarize our conclusions by observing that we have a short exact sequence 0 ⟶ a_{2}M ⟶ M ⟶^{·a_{1}} a_{1}M ⟶ 0, in which a_{2}M is a locally free O_{X_{1}}-module of rank one, and a_{1}M is a locally free O_{X_{2}}-module of rank one. An elementary application of Nakayama’s lemma now implies that M is locally free of rank one over O_{X}, and thus that X is Gorenstein in a neighbourhood of x.

The converse implication (that X Gorenstein implies X_{1} and X_{2} 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 hold in almost all cases. In order to state his result precisely, we recall that if 𝔪 is a maximal ideal of 𝕋 of residue characteristic ℓ, then we say that 𝔪 is ordinary if T_{ℓ} does not lie in 𝔪; otherwise we say that 𝔪 is supersingular. We remark that if 𝔪 is of the second or third type (that is, induces a non-unit maximal ideal of 𝕋^{Eis}) then T_{ℓ} ≡ 1 + ℓ ≡ 1 (mod 𝔪), and so 𝔪 is necessarily ordinary.

Theorem 1.14. The Hecke algebra 𝕋_{𝔪}^{0} is Gorenstein, except possibly in the case that 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 𝕋_{𝔪}^{0} is Gorenstein, except in the case excluded in the statement of the theorem. If 𝔪 is of the second type, then 𝕋_{𝔪}^{0} = 0 is Gorenstein. If 𝔪 is Eisenstein, then [15, cor. II.16.3] shows that 𝕋_{𝔪}^{0} 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 ⟶ 𝕋^{Eis}. Then the quotient M^{0}/I is isomorphic to 𝕋^{0}/I; in particular, it is of order n.

Proof. We first note that 𝕋^{0}/I is isomorphic to the cokernel of (1.4), and so, by part (iii) of proposition 1.8, is cyclic of order n, while the quotient M^{0}/I is isomorphic to M^{0} ⊗_{𝕋^{0}} (𝕋^{0}/I). Thus both 𝕋^{0}/I and M^{0}/I are 𝕋^{0}-modules of finite order annihilated by I, and hence to prove that they are isomorphic, it suffices to prove that their completions at all the maximal ideals 𝔪 of 𝕋^{0} containing I𝕋^{0} are isomorphic. These are precisely the images in 𝕋^{0} of the Eisenstein maximal ideals of 𝕋, and we must thus show that M_{𝔪}^{0}/I and 𝕋_{𝔪}^{0}/I are isomorphic as 𝕋_{𝔪}^{0}-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 M_{𝔪}^{0} and 𝕋_{𝔪}^{0} are already isomorphic as 𝕋_{𝔪}^{0}-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)  Hom_{A}(U ⊗_{B} V, W) ⟶^{∼} Hom_{B}(V, Hom_{A}(U, W)),

defined by sending a morphism ϕ in the source to the morphism v ↦ (u ↦ ϕ(u⊗v)) in the target. For the second isomorphism, let W be a B-module: then we have an isomorphism

(2.2)  Hom_{A}(W, A) ⟶^{∼} Hom_{B}(W, Hom_{A}(B, A)),

defined by sending a morphism ϕ in the source to the morphism w ↦ (b ↦ ϕ(bw)) in the target. (Given the preceding descriptions of these maps, it is immediate to check that both are indeed isomorphisms.)

Let us now fix a pair U,V of B-modules. We have the following diagram:

(2.3) \xymatrix{Hom_{A}(U ⊗_{B} V, A) \ar[r]^{∼} \ar[d]^{∼} & Hom_{B}(V, Hom_{A}(U, A)) \ar[d]^{∼} \\ Hom_{B}(U ⊗_{B} V, Hom_{A}(B, A)) \ar[r]^{∼} & Hom_{B}(V, Hom_{B}(U, Hom_{A}(B, A))).}

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 Hom_{A}(B, A). One easily checks that (2.3) commutes.

By the definition of tensor product, the elements of Hom_{A}(U ⊗_{B} V, A) correspond to B-bilinear pairings ⟨ , ⟩ : U × V ⟶ A. Suppose given such a pairing. The corresponding element ϕ_{A} ∈ Hom_{A}(U ⊗_{B} V, A) gives rise via (2.3) to morphisms ϕ_{B} ∈ Hom_{B}(U ⊗_{B} V, Hom_{A}(B, A)), χ_{A} ∈ Hom_{B}(V, Hom_{A}(U, A)), and χ_{B} ∈ Hom_{B}(V, Hom_{B}(U, Hom_{A}(B, A))).

For the remainder of this section we keep ourselves in the above situation, and in addition assume as a running hypothesis that B, U and V are all locally free of finite rank as A-modules. The first result that we prove will give necessary and sufficient conditions for the morphism ϕ_{B} to be surjective, under the assumption that the pairing ⟨ , ⟩ is perfect.

Lemma 2.4. In the setting described above, suppose that the pairing ⟨ , ⟩ : U × V ⟶ A is a perfect B-bilinear pairing of locally free A-modules of finite rank (that is, suppose that the morphism χ_{A} is an isomorphism of B-modules). Then the morphism ϕ_{B} : U ⊗_{B} V ⟶ Hom_{A}(B, A) is surjective if and only if U/𝔪 is a faithful B/𝔪-module for each maximal ideal 𝔪 of A.

Proof. Since B is finitely generated over A, the morphism ϕ_{B} is surjective if and only if the corresponding morphism ϕ_{B}/𝔪 : U/𝔪 ⊗_{B/𝔪} V/𝔪 ⟶ Hom_{A}(B, A)/𝔪 obtained by tensoring ϕ_{B} 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

Hom_{A}(B, A)/𝔪 ⟶^{∼} Hom_{A/𝔪}(B/𝔪, A/𝔪).

Composing ϕ_{B}/𝔪 with this isomorphism we obtain a morphism ϕ_{B/𝔪} : U/𝔪 ⊗_{B/𝔪} V/𝔪 ⟶ Hom_{A/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 by V/𝔪) to the morphism ϕ_{A/𝔪} = ϕ_{A}/𝔪 : U/𝔪 ⊗_{B/𝔪} V/𝔪 ⟶ A/𝔪 obtained by tensoring ϕ_{A} through with A/𝔪 over A. Since U is locally free over A, there is similarly an isomorphism Hom_{A}(U, A)/𝔪 ⟶^{∼} Hom_{A/𝔪}(U/𝔪, A/𝔪), and composing this with the morphism χ_{A}/𝔪 : V/𝔪 ⟶ Hom_{A}(U, A)/𝔪 we obtain a morphism χ_{A/𝔪} : V/𝔪 ⟶ Hom_{A/𝔪}(U/𝔪, A/𝔪), which also corresponds via (2.3) (again with A replaced by A/𝔪, and so on) to the morphism ϕ_{A/𝔪}. Since χ_{A} is assumed to be an isomorphism, we see that the same is true of χ_{A/𝔪}. Putting these observations together with that of the preceding paragraph, we see that it suffices to prove the lemma with A replaced by A/𝔪, B by B/𝔪, U by U/𝔪 and V by V/𝔪. Thus for the rest of the proof we assume that A is a field.

By assumption χ_{A} induces an isomorphism V ⟶^{∼} Hom_{A}(U, A) of B-modules, and so it suffices to prove the lemma in the case when V = Hom_{A}(U, A) and ϕ_{A} corresponds to the tautological pairing U × Hom_{A}(U, A) ⟶ A. Thus in addition to assuming that A is a field, we assume that V = Hom_{A}(U, A) from now on.

Now ϕ_{B} : U ⊗_{B} Hom_{A}(U, A) ⟶ Hom_{A}(B, A) is surjective if and only if the A-dual morphism B ⟶ Hom_{A}(U ⊗_{B} Hom_{A}(U, A), A) is injective. Composing this morphism with the adjointness isomorphism (2.1) (taking V to be Hom_{A}(U, A) and W to be A), we obtain a morphism

(2.5)  B ⟶ Hom_{B}(Hom_{A}(U, A), Hom_{A}(U, A)),

which is thus injective precisely when ϕ_{B} is surjective. One immediately checks that (2.5) is the canonical morphism describing the B-module structure on Hom_{A}(U, A), and so it is injective precisely when Hom_{A}(U, A) is a faithful B-module. Since Hom_{A}(U, A) is a faithful B-module precisely when U is a faithful B-module, this completes 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 free of 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 End_{B}(U, U) is an isomorphism.

Proof. We begin with some remarks concerning the structure of End_{A}(U, U) and its subring End_{B}(U, U). Since U is free of finite rank over A, the former ring is a matrix ring over A. In particular it is torsion-free and its reduction modulo π acts faithfully on the reduction U/π. More generally, if R is any A-subalgebra of End_{A}(U, U), note that R/π acts faithfully on U/π if and only if R is saturated in End_{A}(U, U).

We let Z denote the centre of End_{B}(U, U). Both Z and End_{B}(U, U) are in particular A-submodules of the matrix ring End_{A}(U, U), and so are free of finite rank as A-modules. Also, they are both saturated in End_{A}(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 End_{B⊗_{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 a matrix ring over B. Since the centre of such a ring is just B itself, we conclude that ι 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 End_{A}(U, U), we see that ι is an isomorphism if and only if ι(B) is saturated in End_{A}(U, U). The lemma follows once one recalls that ι(B) is saturated in End_{A}(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 ϕ_{B}, in situations where it may not be surjective (for example, because the pairing giving rise to ϕ_{A} 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 B_{𝔫}, where 𝔫 ranges over the (finitely many) maximal ideals of B. For any such maximal ideal 𝔫, we let U_{𝔫} and V_{𝔫} denote the completions of U and V at 𝔫. Then U and V also decompose as the direct sums of 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 U_{𝔫} ⊗_{B_{𝔫}} V_{𝔫} ⟶ Hom_{A}(B_{𝔫}, A) induced by ϕ_{B} and V_{𝔫} ⟶ Hom_{A}(U_{𝔫}, A) induced by χ_{A} are both surjective;

(ii) the B_{𝔫}-module U_{𝔫} is free of rank one.

Then the cokernel of the morphism ϕ_{B} is isomorphic to the cokernel of the morphism χ_{A}.

Proof. Writing B ⟶^{∼} ⊕_{𝔫}B_{𝔫}, U ⟶^{∼} ⊕_{𝔫}U_{𝔫}, and V ⟶^{∼} ⊕_{𝔫}V_{𝔫}, we see that the cokernels of ϕ_{B} and χ_{A} also decompose as such direct sums, and so it suffices to verify the lemma after replacing B by one of the B_{𝔫} and U and V by the corresponding U_{𝔫} and V_{𝔫}. Thus we may assume that either ϕ_{B} and χ_{A} are surjections, or that U is a free B-module of rank one.

In the first case, both ϕ_{B} and χ_{A} have trivial cokernels, and so the lemma holds in this case.

In the second case, note that the morphism χ_{B} is equal to the composition of χ_{A} with the isomorphism Hom_{A}(U, A) ⟶^{∼} Hom_{B}(U, Hom_{A}(B, A)). Thus the cokernel of χ_{A} is isomorphic to the cokernel of χ_{B}. In addition, the morphism χ_{B} can be thought of as being obtained from the morphism ϕ_{B} by tensoring over B with the B-dual of the B-module U (which we are assuming to be free of rank one). Thus the cokernel of ϕ_{B} is isomorphic to the cokernel of χ_{B}, and so to the cokernel of χ_{A}, 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 tautological maps of B-modules χ_{A} : V → Hom_{A}(U, A) and ϕ_{B} : U ⊗_{B} V → Hom_{A}(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 U_{𝔫} × V_{𝔫} ⟶ A is perfect, and U_{𝔫}/𝔪 is a faithful A/𝔪-module (here 𝔪 denotes the maximal ideal of A);

(ii) the B_{𝔫}-module U_{𝔫} is free of rank one.

Then the cokernel of the morphism ϕ_{B} is isomorphic to the cokernel of the morphism χ_{A}.

Proof. The pairing U_{𝔫} × V_{𝔫} ⟶ A is perfect if and only if the morphism V_{𝔫} ⟶ Hom_{A}(U_{𝔫}, A) induced by χ_{A} is an isomorphism. Thus lemma 2.4 (with U, V and B replaced by U_{𝔫}, V_{𝔫} and B_{𝔫}) 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 that lemma. □

3. The 𝕋-module X. Constructions.

Let S denote the set of singular points of the geometric fibre of X_{0}(p) in characteristic p. As in section 0 we write S = {x_{0}, …, x_{g}} (where g is equal to the genus of X_{0}(p)). The set S is in bijection with the set of isomorphism classes of supersingular elliptic curves over \overline{𝔽}_{p}, and for each point x_{i} ∈ S we let E_{i} denote a supersingular curve representing the isomorphism class corresponding to x_{i}.

We let X denote the free ℤ-module on the set S, let deg : X ⟶ ℤ denote the ℤlinear map obtained by sending each x_{i} ∈ S to 1 ∈ ℤ, and let X^{0} denote the kernel of deg. The Hecke correspondences on X_{0}(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 𝕋^{Eis}. That is, the Hecke operator T_{n} acts as multiplication by σ′(n).

(ii) The action of the Hecke correspondences on X^{0} makes X^{0} a faithful 𝕋^{0}- module.

(iii) The action of the Hecke correspondences on X makes X a faithful 𝕋-module.

Proof. Let 𝕋′ denote the polynomial ring over ℤ on the countably many generators T_{n}. The action of the Hecke correspondences on X induces an action of 𝕋′ on X.

Let I′ denote the kernel of the surjection of 𝕋′ onto 𝕋^{Eis}, and let J′ denote the kernel of the surjection of 𝕋′ onto 𝕋^{0}. Then I′ ∩ J′ is the kernel of the surjection 𝕋′ ⟶ 𝕋, since the map 𝕋 ⟶ 𝕋^{Eis} ⊕ 𝕋^{0} is injective. Furthermore, since the cokernel of this map is finite of order n, we see that n𝕋′ ⊂ I′ + J′, and hence that n(I′ ∩ J′) ⊂ I′J′.

If we prove (i) and (ii), then we conclude immediately that Ann_{𝕋′}(X) ⊂ I′ ∩J′; a consideration of the exact sequence 0 ⟶ X^{0} ⟶ X ⟶ ℤ ⟶ 0 shows in addition that I′J′ ⊂ Ann_{𝕋′}(X). Taking into account the inclusion n(I′ ∩ J′) ⊂ I′J′ and the fact that X is ℤ-torsion free, we deduce that I′ ∩ J′ = Ann_{𝕋′}(X). This is the claim of (iii).

For any x_{i} ∈ S, deg(T_{n}x_{i}) = σ′(n). Thus the action of 𝕋′ on ℤ (as it appears in the preceding exact sequence) kills I′, and yields a faithful action of 𝕋′/I′ ⟶^{∼} 𝕋^{Eis} on ℤ. This proves (i).

Let T_{0}(p) denote the torus that is the connected component of the fibre in characteristic p of the Néron model of the Jacobian J_{0}(p) of X_{0}(p). The Hecke correspondences act on T_{0}(p) functorially via their action on J_{0}(p), and hence also act on the character lattice Hom(T_{0}(p)_{/\overline{𝔽}_{p}}, 𝔾_{m/\overline{𝔽}_{p}}) of T_{0}(p). Furthermore (a) there is a canonical isomorphism (of Hecke modules) of X^{0} with this character lattice; (b) we can recover the Hecke action on T_{0}(p) from the Hecke action on its character lattice; (c) the endomorphisms of J_{0}(p) act faithfully on its toric reduction T_{0}(p);

(d) we can recover the Hecke action on cusp forms from the Hecke action on J_{0}(p) by passing to its cotangent space. Thus the annihilator of the 𝕋′-action on X^{0} is equal to annihilator of the 𝕋′-action on the space of weight two cusp forms on Γ_{0}(p), which is J′. 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 the character group of the connected component of the fibre in characteristic p of the Néron model of the generalized Jacobian of the non-proper curve Y_{0}(p) = X_{0}(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 faithfully as an algebra of self-adjoint operators on a finite-dimensional vector space equipped 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 equipped with 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 x_{i} ∈ S we let e_{i} denote one half of the number of automorphisms of E_{i}. The following lemma is pointed out in [6, p. 117]. (Recall that Δ denotes the denominator of (p − 1)/12 when this fraction is written in lowest terms.)

Lemma 3.3. The product ∏_{i=0}^{g} e_{i} is equal to Δ.

Proof. This follows from the well-known explicit calculation of the e_{i}, 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 e_{i} = 1 unless j(E_{i}) = 0 (respectively j(E_{i}) = 1728), in which case e_{i} = 3 (respectively e_{i} = 2); furthermore, this j-value actually is supersingular in characteristic 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 formula

⟨x_{i}, x_{j}⟩ = e_{i}δ_{i,j} = e_{j}δ_{i,j} = one-half the number of isomorphisms from E_{i} to E_{j}.

If we restrict this pairing to the submodule X^{0} of X, we obtain the Picard-Lefschetz 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 E_{i}. There is some overlap 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 the Picard 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 the lattice 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 x_{i} and x_{j} of S, let L_{i,j} denote the quadratic space whose underlying ℤ-module is the space of isogenies from E_{i} to E_{j}, equipped with the quadratic form ϕ ↦ 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 x_{i} and x_{j} of S, L_{i,j} is a free ℤ-module of rank 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 L_{i,j} as a ℤ-module is equal to the rank of L_{i,i} for any choice of 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. □

Lemma 3.7. For any two elements x_{i} and x_{j} of S there is a natural isomorphism of quadratic spaces L_{i,j} ⟶^{∼} L_{j,i}.

Proof. The required isomorphism is obtained by sending any isogeny from E_{i} to E_{j}, of some degree n say, to its dual isogeny, which maps from E_{j} to E_{i}, and is also of degree n. □

If n is any natural number, let r_{n}(L_{i,j}) denote the number of elements of L_{i,j} of degree n.

Lemma 3.8. If x_{i} and x_{j} are any two elements of S, then

⟨T_{n}x_{i}, x_{j}⟩ = one-half the number of degree n isogenies from E_{i} to E_{j}

= \frac{1}{2}r_{n}(L_{i,j}).

Proof. Since T_{n}x_{i} is the formal sum of all curves that are the image of E_{i} under an n-isogeny, and since ⟨x_{k}, x_{j}⟩ equals one half the number of isomorphisms from E_{k} to E_{j}, for any index k, we see that ⟨T_{n}x_{i}, x_{j}⟩ does indeed equal one-half the number of degree n isogenies from E_{i} to E_{j}, 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, ⟨T x, y⟩ = ⟨x, T y⟩.

Proof. It suffices to verify this when x = x_{i} and y = y_{j} are elements of S and T = T_{n} for some natural number n. In this case lemmas 3.7 and 3.8 show that

⟨T_{n}x_{i}, x_{j}⟩ = \frac{1}{2}r_{n}(L_{i,j}) = \frac{1}{2}r_{n}(L_{j,i}) = ⟨T_{n}x_{j}, x_{i}⟩ = ⟨x_{i}, T_{n}x_{j}⟩. □

Corollary 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 ϕ_{A} in section 2) and a corresponding morphism of 𝕋-modules

(3.10)  X ⊗_{𝕋} X ⟶ Hom_{ℤ}(𝕋, ℤ)

(the morphism denoted ϕ_{B} in section 2). Composing (3.10) with the isomorphism of proposition 1.3 (ii) we obtain a morphism θ : X ⊗_{𝕋} X ⟶ N . From the definition of the pairing of proposition 1.3 and the construction of (3.10) we see that the map θ is determined by the formula a_{n}(θ(x ⊗ x′)) = ⟨T_{n}x, x′⟩, for x, x′ ∈ 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 x_{i} and x_{j} are any two elements of of S, then for any n ≥ 1,

we have a_{n}(θ(x_{i} ⊗ x_{j})) = \frac{1}{2}r_{n}(L_{i,j}).

Proof. Combining the above discussion with lemma 3.8 we see that if n ≥ 1 then a_{n}(θ(x_{i} ⊗ x_{j})) = ⟨T_{n}x_{i}, x_{j}⟩ = \frac{1}{2}r_{n}(L_{i,j}). □

Definition 3.12. Let x denote the element of X defined by x = Δ ∑_{i=0}^{g} \frac{x_{i}}{e_{i}} . (Note

that lemma 3.3 implies that this is an element of X.)

This is the element denoted by e_{0} in [6, (4.8)]. Observe that ⟨x, x_{i}⟩ = Δ for each element x_{i} of S. Thus ⟨x, x⟩ = Δ deg(x), for any x ∈ X.

Let X^{Eis} denote the maximal 𝕋-submodule of X on which the action of 𝕋 factors through 𝕋^{Eis}. The short exact sequence 0 ⟶ X^{0} ⟶ X ⟶ ℤ ⟶ 0 along with parts (i) and (ii) of theorem 3.1 shows that X^{Eis} must be a rank one ℤ-submodule of X.

Lemma 3.13. X^{Eis} is the free ℤ-module of rank one generated by x.

Proof. For any element x_{j} of S, we compute that

⟨T_{n}x, x_{j}⟩ = ⟨x, T_{n}x_{j}⟩ = Δ ∑_{i} \frac{⟨x_{i}, T_{n}x_{j}⟩}{e_{i}}

= Δ ∑_{i} number of n-isogenies with source E_{j} and target isomorphic to E_{i}

= Δ × number of n-isogenies with source E_{j} = Δσ′(n).

Thus the coefficient of x_{j} in T_{n}x must be Δσ′(n)/e_{j}. Since j was arbitrary, we find that T_{n}x = σ′(n)x. Thus x lies in X^{Eis}.

Now X^{Eis} is free of rank one over ℤ, and the coefficients Δ/e_{i} of the generators x_{i} of X in x are coprime elements of ℤ (as follows from lemma 3.3 and the explicit values of the numbers e_{i} which are recalled in the course of its proof). Thus we see that x is not divisible by any non-unit element of ℤ in X, and so it must generate X^{Eis}. □

Proposition 3.14. Let x_{i} be any element of S. Then θ(x ⊗ x_{i}) is equal to the modular form ΔE.

Proof. Similarly to the proof of proposition 3.11, we compute

a_{n}(θ(x ⊗ x_{i})) = ⟨T_{n}x, x_{i}⟩ = ⟨σ′(n)x, x_{i}⟩ = Δσ′(n),

since ⟨x, x_{i}⟩ = Δ for each generator x_{i} of X. □

Proposition 3.15. The following statements hold:

(i) if x_{i} and x_{j} are any two elements of S, then a_{0}(θ(x_{i} ⊗ x_{j})) = \frac{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;

(iii) if x_{i} and x_{j} are any two elements of S, then θ(x_{i} ⊗ x_{j}) = \frac{1}{2}Θ(L_{i,j});

(iv) a_{0}(E) = ∑_{i} \frac{1}{2e_{i}};

(v) deg x = n;

(vi) ∑_{i} \frac{1}{e_{i}} = \frac{p − 1}{12}.

Proof. We begin by proving that the six statements are mutually equivalent. Since the formation of a_{0}(θ(x ⊗ y)) and (deg(x) deg(y))/2 are both bilinear in x and y, statements (i) and (ii) are clearly equivalent. Proposition 3.11 together with the fact that a_{0}(Θ(L_{i,j})) = 1 shows that statements (i) and (iii) are equivalent. Since deg(x) = Δ ∑_{i} \frac{1}{e_{i}} and a_{0}(E) = (p−1)/24 = n/2Δ, we see that statements (iv), (v)

and (vi) are equivalent.

We now show that statements (i) and (iv) are equivalent. To this end, we observe that

a_{0}(θ(x_{i} ⊗ x_{j})) = \frac{1}{deg(x)} (a_{0}(θ(x ⊗ x_{j})) + a_{0}(θ((deg(x)x_{i} − x) ⊗ x_{j}))) .

Note that deg(x)x_{i} − x lies in X^{0}, which implies that θ((deg(x)x_{i} − x) ⊗ x_{j}) lies in M^{0}, and so has vanishing constant term. Proposition 3.14 asserts that θ(x ⊗ x_{i}) = ΔE. Thus we find that

a_{0}(E) = \frac{deg(x)a_{0}(θ(x_{i} ⊗ x_{j}))}{Δ} = a_{0}(θ(x_{i} ⊗ x_{j})) ∑_{i} \frac{1}{e_{i}} ,

which 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(L_{i,j}) are weight two modular forms on Γ_{0}(p). The formula of proposition 3.11 then shows

that θ(x_{i} ⊗ x_{j}) and \frac{1}{2}Θ(L_{i,j}) coincide up to their constant terms, and so coincide.

Thus a_{0}(θ(x_{i} ⊗ x_{j})) = \frac{1}{2}a_{0}(Θ(L_{i,j})) = \frac{1}{2}, proving part (i) of proposition 3.15. 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 modularity of the theta series Θ(L_{i,j}). Indeed, the construction of the map θ is purely algebrogeometric, and proposition 3.11 then shows that Θ(L_{i,j}) is a modular form, provided we allow ourselves to modify its constant term as necessary. Proposition 3.15 shows that in fact no such modification is necessary. To what extent does the proof of this 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 no reference 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 Eisenstein series E. One can easily give a purely algebro-geometric proof that there exists a Hecke eigenform E ∈ N on which 𝕋 acts through 𝕋^{Eis}, but one is left with the problem of determining a_{0}(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 of this 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 constant term of the modular forms θ(x_{i} ⊗ x_{j}). (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 ⟶ Hom_{ℤ}(X , ℤ) induced by the pairing of definition 3.4 is isomorphic to ℤ/Δ, regarded as a 𝕋-module via the surjection 𝕋 ⟶ 𝕋^{Eis}. In particular the discriminant of this pairing is equal to Δ.

(ii) The cokernel of the morphism X^{0} ⟶ Hom_{ℤ}(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 𝕋-module via the surjection 𝕋 ⟶ 𝕋^{Eis}. In particular the discriminant of this pairing 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 Δ, by lemma 3.3.

Since the pairing of definition 3.4 has a non-zero discriminant, it induces an embedding X ⟶ Hom_{ℤ}(X , ℤ), which is 𝕋-linear, by corollary 3.9. Since ⟨x, x⟩ = Δ deg(x) for all x ∈ X , we see that under this map the element x of X maps to the element Δ deg of Hom_{ℤ}(X , ℤ). Thus we obtain a commutative diagram of morphisms of 𝕋-modules

(3.17)  \xymatrix{0 \ar[r] & X^{Eis} \ar[r] \ar[dd]^{(1)} & X^{Eis} ⊕ X^{0} \ar[r] \ar[d]^{(2a)} & X^{0} \ar[r] \ar[dd]^{(3)} & 0 \\  &  & X \ar[d]^{(2b)} \\ 0 \ar[r] & ℤ deg \ar[r] & Hom_{ℤ}(X , ℤ) \ar[r] & Hom_{ℤ}(X^{0}, ℤ) \ar[r] & 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/Δ, regarded as a 𝕋-module via the surjection 𝕋 ⟶ 𝕋^{Eis} (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 ⟶ 𝕋^{Eis} (as follows from part (i) of theorem 3.1 and part (v) of proposition 3.15), and (2b) has cokernel of order Δ (as follows from the discriminant calculation made at the beginning of this proof). Since Δ 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} ⟶ Hom_{ℤ}(X^{0}, ℤ) is isomorphic to the group of connected components of the geometric fibre at p of the Néron model of J_{0}(p), which is a cyclic group of order n on which 𝕋 acts through its quotient 𝕋^{Eis}, 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]), and so 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 3.22 of [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 ℓ be a prime. The tensor product 𝕋 ⊗_{ℤ} ℤ_{ℓ} is a direct sum of the complete local rings 𝕋_{𝔪}, where 𝔪 ranges over the maximal ideals of 𝕋 of residue characteristic ℓ. Thus the tensor product X ⊗_{ℤ} ℤ_{ℓ} decomposes as a direct sum of its completions X_{𝔪}. We may tensor the pairing of definition 3.4 by ℤ_{ℓ} over ℤ to obtain a pairing

(3.18)  X ⊗_{ℤ} ℤ_{ℓ} × X ⊗_{ℤ} ℤ_{ℓ} ⟶ ℤ_{ℓ}.

Since this pairing is 𝕋 ⊗_{ℤ} ℤ_{ℓ}-bilinear (by corollary 3.9) we see that for distinct maximal ideals 𝔪 and 𝔪′ of residue characteristic ℓ, this pairing becomes trivial when restricted to X_{𝔪} × X_{𝔪′}. Thus this pairing is the direct sum of its restrictions X_{𝔪} × X_{𝔪} ⟶ ℤ_{ℓ}, as 𝔪 ranges over the maximal ideals of 𝕋 having residue characteristic ℓ.

Similarly, we can restrict (3.18) to obtain the pairing

(3.19)  X^{0} ⊗_{ℤ} ℤ_{ℓ} × X^{0} ⊗_{ℤ} ℤ_{ℓ} ⟶ ℤ_{ℓ},

and we similarly observe that it decomposes as the direct sum of its restrictions X_{𝔪}^{0} × X_{𝔪}^{0} ⟶ ℤ_{ℓ}.

Corollary 3.20. Let 𝔪 be a maximal ideal of 𝕋 of residue characteristic ℓ.

(i) If ℓ does not divide Δ then the pairing X_{𝔪}×X_{𝔪} ⟶ ℤ_{ℓ} is a perfect 𝕋_{𝔪}-bilinear pairing of free ℤ_{ℓ}-modules.

(ii) If 𝔪 is not an Eisenstein prime, then the pairing X_{𝔪}^{0} ×X_{𝔪}^{0} ⟶ ℤ_{ℓ} is a perfect 𝕋_{𝔪}^{0}-bilinear pairing of free ℤ_{ℓ}-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 ℓ does not divide Δ, then part (ii) follows from part (i) and the direct sum decomposition X_{𝔪}^{Eis} ⊕X_{𝔪}^{0} ⟶^{∼} X_{𝔪}, together with the fact that X^{Eis} and X^{0} are orthogonal under the pairing (by 𝕋-bilinearity). On the other hand, if ℓ does not divide n, then part (ii) follows from the discriminant calculation of lemma 3.16 (ii). Since Δ and n are coprime by construction, we see that 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 the proof of lemma 3.16.)

The final observation that we will make in this section is that the results already established suffice to recover Eichler’s theorem [5] that the space of weight two modular forms on Γ_{0}(p) is spanned by theta series.

Proposition 3.21. The morphism X ⊗_{ℤ} ℚ ⊗_{𝕋⊗_{ℤ}ℚ} X ⊗_{ℤ} ℚ ⟶ M ⊗_{ℤ} ℚ induced by (0.2) is an isomorphism. In particular, the ℚ-vector space M ⊗_{ℤ} ℚ is spanned by the theta series Θ(L_{i,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, we find that it induces a perfect pairing U × U ⟶ A, and so we are in the situation of lemma 2.4. It follows from that lemma, together with corollary 3.2, that the morphism ϕ_{B} (in the notation of that lemma) is an isomorphism. However, ϕ_{B} 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 ⟶^{∼} Hom_{ℚ}(𝕋⊗_{ℤ}ℚ, ℚ)). The proposition now follows by observing that proposition 3.11 implies that θ tensored with 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 𝕋_{𝔪}-modules X_{𝔪} and X_{𝔪}^{0} (as 𝔪 ranges over the maximal ideals of 𝕋), and their relation to the arithmetic of the Jacobian J_{0}(p).

We begin by studying the case when 𝔪 is an Eisenstein maximal ideal of 𝕋 of residue characteristic ℓ. As in the proof of theorem 1.10, we let I denote the kernel of the map 𝕋_{𝔪} ⟶ 𝕋_{𝔪}^{Eis} ⟶^{∼} ℤ_{ℓ} and J denote the kernel of the map 𝕋_{𝔪} ⟶ 𝕋_{𝔪}^{0}. Then I and J are each principal (since their intersection is zero and each induces a Cartier divisor in the closed subscheme cut out by the other – see the proofs of theorem 1.10 and lemma 1.11), and so I + J is generated by a pair of elements η ∈ I, γ ∈ J. Now γ is a lifting of the generator of the image of J in ℤ_{ℓ}, and so is congruent to some power ℓ^{n} of ℓ modulo I. Thus we have I + J = (η, ℓ^{n})𝕋_{𝔪}, and 𝕋_{𝔪}/(I + J) ⟶^{∼} ℤ_{ℓ}/ℓ^{n}. We conclude that 𝔪 = (η, ℓ)𝕋_{𝔪}.

Lemma 4.1. Let 𝔪 be an Eisenstein maximal ideal of residue characteristic ℓ.

(i) If N is any finitely generated 𝕋_{𝔪}/ℓ-module, then N[𝔪] (the 𝕋-submodule of 𝔪-torsion points of N) and N/𝔪 are 𝕋/𝔪-vector spaces of equal rank.

(ii) If 0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0 is a short exact sequence of 𝕋_{𝔪}/ℓ-modules, there is an induced long exact sequence

0 ⟶ N′[𝔪] ⟶ N[𝔪] ⟶ N′′[𝔪] ⟶ N′/𝔪 ⟶ N/𝔪 ⟶ N′′/𝔪 ⟶ 0.

Proof. To prove (i), note that N[η] and N/η are 𝕋_{𝔪}-modules annihilated by both ℓ and η, and hence by 𝔪, so that N[η] = N[𝔪] and N/η = N/𝔪. That they are of equal rank as 𝕋/𝔪-vector spaces (equivalently, of equal length as 𝕋_{𝔪}-modules) follows from the exact sequence 0 ⟶ N[η] ⟶ N ⟶^{·η} N ⟶ N/η ⟶ 0 and the additivity of lengths in exact sequences.

To prove (ii), tensor the given exact sequence with the complex 𝕋_{𝔪}/ℓ ⟶^{·η} 𝕋_{𝔪}/ℓ of free 𝕋_{𝔪}/ℓ-modules and apply the snake lemma. This yields the long exact sequence

0 ⟶ N′[η] ⟶ N[η] ⟶ N′′[η] ⟶ N′/η ⟶ N/η ⟶ N′′/η ⟶ 0.

Since ℓ and η generate the maximal ideal 𝔪𝕋_{𝔪}, we can rewrite this as

0 ⟶ N′[𝔪] ⟶ N[𝔪] ⟶ N′′[𝔪] ⟶ N′/𝔪 ⟶ N/𝔪 ⟶ N′′/𝔪 ⟶ 0,

as claimed in the statement of part (ii). □

Theorem 4.2. Let 𝔪 be an Eisenstein maximal ideal of 𝕋. Then X_{𝔪} is a free 𝕋_{𝔪}-module of rank one, and X_{𝔪}^{0} is a free 𝕋_{𝔪}^{0}-module of rank one.

Proof. As above, we let ℓ denote the residue characteristic of 𝔪. From corollary 3.2 we see that X_{𝔪}⊗_{ℤ_{ℓ}}ℚ_{ℓ} is free of rank one over 𝕋_{𝔪}⊗_{ℤ_{ℓ}}ℚ_{ℓ}, and that similarly X_{𝔪}^{0}⊗_{ℤ_{ℓ}}ℚ_{ℓ} is free of rank one over 𝕋_{𝔪}^{0} ⊗_{ℤ_{ℓ}} ℚ_{ℓ}. Thus to prove the theorem it suffices to show that each of the 𝕋/𝔪 ⟶^{∼} 𝕋^{0}/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-modules

0 ⟶ X^{0} ⟶ X ⟶^{deg} ℤ ⟶ 0

with the flat 𝕋-module 𝕋_{𝔪} to obtain the short exact sequence of 𝕋_{𝔪}-modules

0 ⟶ X_{𝔪}^{0} ⟶ X_{𝔪} ⟶^{deg} ℤ_{ℓ} ⟶ 0.

This is a short exact sequence of torsion-free (equals flat) ℤ_{ℓ}-modules, and so reducing modulo ℓ yields the short exact sequence 0 ⟶ X_{𝔪}^{0}/ℓ ⟶ X_{𝔪}/ℓ ⟶^{deg} ℤ/ℓ ⟶ 0, giving rise, by part (ii) of lemma 4.1, to the long exact sequence

(4.3) 0 ⟶ X_{𝔪}^{0}/ℓ [𝔪] ⟶ X_{𝔪}/ℓ [𝔪] ⟶^{deg} ℤ/ℓ

⟶ X^{0}/𝔪 ⟶ X /𝔪 ⟶^{deg} ℤ/ℓ ⟶ 0.

(Note that 𝕋 acts on the quotient ℤ ⟶^{∼} X /X^{0} through 𝕋^{Eis}, so that ℤ/ℓ [𝔪] ⟶^{∼} (ℤ/ℓ)/𝔪 ⟶^{∼} ℤ/ℓ.) 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 /ℓ [𝔪] and X^{0}/ℓ [𝔪] are of the same rank. A consideration of (4.3) shows that it suffices to prove that the image of X /ℓ [𝔪] in ℤ/ℓ is trivial.

Let x̄ ∈ X_{𝔪}/ℓ [𝔪]. We can find an element x ∈ X that reduces to x̄ modulo ℓ. Let x_{i} be an element of S. Write f = θ(x ⊗ x_{i}) ∈ M. Then f − a_{1}(f)E is an element of ℓN, since x is annihilated by 𝔪 modulo ℓ, and so (in the notation of proposition 1.1) A_{0}(f) ≡ A_{0}(a_{1}(f)E) = a_{1}(f)n mod ℓ. By proposition 3.15 we see that A_{0}(f) = 2Δ deg(x)/2 = Δ deg(x). Thus we conclude that Δ deg(x) ≡ a_{1}(f)n mod ℓ. Since 𝔪 is an Eisenstein prime, proposition 1.7 shows that ℓ divides n, and so is coprime to Δ. Thus deg(x) ≡ 0 mod ℓ, and so x̄ does have trivial image in ℤ/ℓ. 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 over 𝕋^{0}/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 J_{0}(p) of X_{0}(p), and let T_{0}(p) denote the connected component of the characteristic p fibre of J. Then T_{0}(p) is a torus over 𝔽_{p}. Recall that X^{0} can be naturally identified with the character lattice Hom_{\overline{𝔽}_{p}}(T_{0}(p), 𝔾_{m}) of T_{0}(p). Since T_{0}(p) is defined over 𝔽_{p}, we see that X^{0} is equipped with an action of Gal(\overline{𝔽}_{p}/𝔽_{p}), and that T_{0}(p) can in turn be recovered from X^{0} equipped with its action of Gal(\overline{𝔽}_{p}/𝔽_{p}) as Hom(X^{0}, 𝔾_{m}) (at first defined over \overline{𝔽}_{p}, and then descended to 𝔽_{p} using the descent data induced by the Galois action on X^{0}). By regarding the Gal(\overline{𝔽}_{p}/𝔽_{p})-action on X^{0} as being an unramified action of Gal(\overline{ℚ}_{p}/ℚ_{p}), we see that the expression Hom(X^{0}, 𝔾_{m}) equally well defines a torus T over ℤ_{p} that has T_{0}(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 the formal completion of J along its characteristic p fibre, and so the group scheme T [ℓ] embeds as a sub-group scheme of J [ℓ] (at first formally, but then actually, since it is finite). Passing to characteristic zero fibres shows that T [ℓ]_{/ℚ_{p}} embeds into J_{0}(p)[ℓ]_{/ℚ_{p}}. Now T [ℓ] is Cartier dual to X^{0}/ℓ, and the cokernel of the preceding embedding is naturally identified with X^{0}/ℓ. We thus obtain a short exact sequence of Gal(\overline{ℚ}_{p}/ℚ_{p})-modules

0 ⟶ Hom(X^{0}/ℓ, μ_{ℓ}) ⟶ J_{0}(p)[ℓ](\overline{ℚ}_{p}) ⟶ X^{0}/ℓ ⟶ 0,

where Hom(X^{0}/ℓ, μ_{ℓ}) is naturally identified with T [ℓ](\overline{ℚ}_{p}). (This is the short exact sequence [23, (2.4)].)

Apply Hom_{𝕋^{0}}(𝕋^{0}/𝔪, –) to this short exact sequence, to obtain the exact sequence

0 ⟶ Hom(X^{0}/𝔪, μ_{ℓ}) ⟶ J_{0}(p)[𝔪](\overline{ℚ}_{p}) ⟶ X^{0}/ℓ [𝔪],

where J_{0}(p)[𝔪] denotes the finite subgroup scheme of J_{0}(p)[ℓ] that is the simultaneous kernel of all element of 𝔪. Observe that [15, p. 99 and cor. II.16.3] imply that J_{0}(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 ℤ_{p}, 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 discussion of the preceding paragraph that the specialization to characteristic p of the points of Hom(X^{0}/𝔪, μ_{ℓ}) lie in T_{0}(p)[ℓ], and so have trivial image in this connected component group. Thus Hom(X^{0}/𝔪, μ_{ℓ}) must be only one-dimensional over 𝕋/𝔪. The theorem now follows. □

We now turn to an analysis of X_{𝔪} and X_{𝔪}^{0} when 𝔪 is of the first type (so that

𝕋_{𝔪} ⟶^{∼} 𝕋_{𝔪}^{0} and X_{𝔪}^{0} ⟶^{∼} X_{𝔪}.)

Lemma 4.4. Let 𝔪 be a maximal ideal of 𝕋 of the first type, having residue characteristic ℓ, and let W be a finite length 𝕋_{𝔪}^{0}/ℓ-module. Then the 𝕋_{𝔪}^{0}/ℓ-modules X_{𝔪}^{0}/ℓ ⊗_{𝕋_{𝔪}^{0}/ℓ} W and Hom_{𝕋_{𝔪}^{0}/ℓ}(W, X_{𝔪}^{0}/ℓ) are naturally ℤ/ℓ-dual to one another. In particular, they are of equal dimension as ℤ/ℓ-vector spaces.

Proof. Since 𝔪 is non-Eisenstein, it follows from corollary 3.20 (ii) that the pairing (3.19) induces a perfect 𝕋_{𝔪}^{0}/ℓ-bilinear pairing X_{𝔪}^{0}/ℓ × X_{𝔪}^{0}/ℓ ⟶ ℤ/ℓ. Using this we compute (using the adjointness between Hom and ⊗) that

Hom_{ℤ/ℓ}(X_{𝔪}^{0}/ℓ ⊗_{𝕋_{𝔪}^{0}/ℓ} W, ℤ/ℓ) ⟶^{∼} Hom_{𝕋_{𝔪}^{0}/ℓ}(W, Hom_{ℤ/ℓ}(X_{𝔪}^{0}/ℓ, ℤ_{ℓ}))

⟶^{∼} Hom_{𝕋_{𝔪}^{0}/ℓ}(W, X_{𝔪}^{0}/ℓ).

Since a ℤ/ℓ-vector space and its dual have the same dimension, this proves the lemma. □

For the remainder of this section we fix an embedding \overline{ℚ} → \overline{ℚ}_{p} and thus obtain an embedding of Galois groups Gal(\overline{ℚ}_{p}/ℚ_{p}) → Gal(\overline{ℚ}/ℚ). (In other words, we fix a choice of decomposition group at p in Gal(\overline{ℚ}/ℚ).) The following simple lemma will be required in the proof of theorem 4.6 below. We refer to [24, p. 189] for the notion of finiteness of a Gal(\overline{ℚ}_{p}/ℚ_{p})-representation.

Lemma 4.5. Let A be a finite local ring with residue field k, let ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(k) be an irreducible two-dimensional representation of Gal(\overline{ℚ}/ℚ) over k, let U be a finite-length A-module equipped with an action of Gal(\overline{ℚ}/ℚ), and let V be a finite-length A-module equipped with an action of Gal(\overline{ℚ}_{p}/ℚ_{p}). Suppose that

(i) ρ is not finite at p;

(ii) the semi-simplification of U as an A[Gal(\overline{ℚ}/ℚ)]-module is isomorphic to a direct sum of copies of ρ;

(iii) V is a finite Gal(\overline{ℚ}_{p}/ℚ_{p})-module;

(iv) there is a Gal(\overline{ℚ}_{p}/ℚ_{p})-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(\overline{ℚ}/ℚ)]-modules such that the successive quotients F^{i}/F^{i+1} are isomorphic to ρ (i = 0, …, l − 1). Let G^{i} = V ∩ F^{i}. Then G^{i}/G^{i+1} embeds into F^{i}/F^{i+1}. Since the latter k-vector space is two-dimensional and not finite as a Gal(\overline{ℚ}_{p}/ℚ_{p})- module, while G^{i}/G^{i+1} is finite (being a subquotient of the finite representation V), we see that G^{i}/G^{i+1} is at most one-dimensional. Thus lg U = ∑_{i=0}^{l−1} lg F^{i}/F^{i+1} ≥ 2 ∑_{i=0}^{l−1} lg G^{i}/G^{i+1} = 2 lg V. □

If 𝔪 is a maximal ideal of the first type then attached to 𝔪 there is a twodimensional irreducible representation ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) ⟶ GL_{2}(𝕋^{0}/𝔪), characterized by the property that it is unramified away from p and ℓ, and that for any prime q distinct from p and ℓ, the characteristic polynomial of ρ(Frob_{q}) is equal to X^{2} − T_{q}X + q (mod 𝔪) [15, prop. II.14.1]. Furthermore, the 𝕋/𝔪 [Gal(\overline{ℚ}/ℚ)]- module J_{0}[𝔪](\overline{ℚ}) has a Jordan-Hölder filtration whose successive irreducible quotients are isomorphic to the representation ρ_{𝔪} [15, proof of prop. II.14.2].

Note that the set S that forms a basis for X is equipped with an action of Gal(\overline{𝔽}_{p}/𝔽_{p}) (since it is a set of \overline{𝔽}_{p}-valued points of a curve defined over 𝔽_{p}). Thus X is naturally a Gal(\overline{𝔽}_{p}/𝔽_{p})-module. We may equally well regard X as an unramified Gal(\overline{ℚ}_{p}/ℚ_{p})-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 ℓ. Then there is a short exact sequence of 𝕋_{𝔪}^{0}/ℓ [Gal(\overline{ℚ}_{p}/ℚ_{p})]-modules

0 ⟶ Hom(X_{𝔪}^{0}, μ_{ℓ}) ⟶ J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪} ⟶ X_{𝔪}^{0}/ℓ ⟶ 0,

which is split as a short exact sequence of 𝕋_{𝔪}^{0}/ℓ-modules.

Proof. The theorem is an extension of [23, thm. 2.3], and will follow the same lines of proof as that result. Just as in the proof of theorem 4.2 we obtain a short exact sequence

0 ⟶ Hom(X^{0}/ℓ, μ_{ℓ}) ⟶ J_{0}(p)[ℓ](\overline{ℚ}_{p}) ⟶ X^{0}/ℓ ⟶ 0

of 𝕋^{0}/ℓ [Gal(\overline{ℚ}_{p}/ℚ_{p})]-modules, which we may localize at 𝔪 to yield the required short exact sequence of 𝕋_{𝔪}^{0}/ℓ [Gal(\overline{ℚ}_{p}/ℚ_{p})]-modules

(4.7)  0 ⟶ Hom(X_{𝔪}^{0}/ℓ, μ_{ℓ}) ⟶ J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪} ⟶ X_{𝔪}^{0}/ℓ ⟶ 0.

We will show that (4.7) remains short exact after applying Hom_{𝕋_{𝔪}^{0}/ℓ}(W, –), for any finite length 𝕋_{𝔪}^{0}/ℓ-module W. Taking W = X_{𝔪}^{0}/ℓ will then yield a 𝕋_{𝔪}^{0}/ℓ-linear map s : X_{𝔪}^{0}/ℓ → J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪} which lifts the identity map from X_{𝔪}^{0}/ℓ to itself, and so splits (4.7).

Choose a finite length 𝕋_{𝔪}^{0}/ℓ-module W. Applying Hom_{𝕋_{𝔪}^{0}/ℓ}(W, –) to (4.7) yields an exact sequence of 𝕋_{𝔪}^{0}/ℓ [Gal(\overline{ℚ}_{p}/ℚ_{p})]-modules

(4.8) 0 ⟶ Hom(X_{𝔪}^{0}/ℓ ⊗_{𝕋_{𝔪}^{0}/ℓ} W, μ_{ℓ})

⟶ Hom_{𝕋_{𝔪}^{0}/ℓ}(W, J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪}) ⟶ Hom_{𝕋_{𝔪}^{0}/ℓ}(W, X_{𝔪}^{0}/ℓ).

(Here we regard W as being endowed with the trivial Gal(\overline{ℚ}_{p}/ℚ_{p})-action. We have also used the isomorphism Hom_{𝕋_{𝔪}^{0}/ℓ}(W, Hom(X_{𝔪}^{0}/ℓ, μ_{ℓ})) ⟶^{∼} Hom(X_{𝔪}^{0}/ℓ⊗_{𝕋_{𝔪}^{0}/ℓ} W, μ_{ℓ}) provided by the adjointness between Hom and ⊗.) We must show that (4.8) 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(\overline{ℚ}_{p}/ℚ_{p})-module Hom(X_{𝔪}^{0}/ℓ ⊗_{𝕋_{𝔪}^{0}/ℓ} W, μ_{ℓ}) is finite at p. (To see this, note that if we forget the 𝕋_{𝔪}^{0}/ℓ-action then we obtain a direct sum of copies of μ_{ℓ}.)

(ii) Since J_{0}(p)[ℓ](\overline{ℚ}_{p}) and J_{0}(p)[ℓ](\overline{ℚ}) are of course equal, the Gal(\overline{ℚ}_{p}/ℚ_{p})- action on Hom_{𝕋_{𝔪}^{0}/ℓ}(W, J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪}) = Hom_{𝕋_{𝔪}^{0}/ℓ}(W, J_{0}(p)[ℓ](\overline{ℚ})_{𝔪}) extends to a Gal(\overline{ℚ}/ℚ)-action, and the semi-simplification of Hom_{𝕋_{𝔪}^{0}/ℓ}(W, J_{0}(p)[ℓ](\overline{ℚ})_{𝔪}) as a Gal(\overline{ℚ}/ℚ)-module is a direct sum of copies of ρ_{𝔪}. (This follows from the fact that this Gal(\overline{ℚ}/ℚ)-module is a successive extension of copies of J_{0}(p)[𝔪](\overline{ℚ}), which itself is a successive extension of copies of ρ_{𝔪}, as we observed above.)

Since 𝔪 is non-Eisenstein, we see by [23, prop. 2.2] that ρ_{𝔪} is not finite at p. Taking into account (i) and (ii) above we see that lemma 4.5 implies that the dimension of Hom(X_{𝔪}^{0}/ℓ ⊗_{𝕋_{𝔪}^{0}/ℓ} W, μ_{ℓ}) over ℤ/ℓ is at most one-half that of Hom_{𝕋_{𝔪}^{0}/ℓ}(W, J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪}). A consideration of (4.8) together with lemma 4.4 now shows that each of Hom(X_{𝔪}^{0}/ℓ ⊗_{𝕋_{𝔪}^{0}/ℓ} W, μ_{ℓ}) and Hom_{𝕋_{𝔪}^{0}/ℓ}(W, X_{𝔪}^{0}/ℓ) must be of exactly one-half the dimension of Hom_{𝕋_{𝔪}^{0}/ℓ}(W, J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪}) over ℤ/ℓ, 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 dimension of J_{0}(p)[𝔪] over 𝕋^{0}/𝔪 is equal to twice the dimension of X_{𝔪}^{0}/𝔪 over 𝕋^{0}/𝔪.

Proof. This follows by taking W to be 𝕋^{0}/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 ℓ, then X_{𝔪}^{0}/ℓ is a faithful 𝕋_{𝔪}^{0}/ℓ-module.

Proof. We see by theorem 4.6 that it suffices to show that J_{0}(p)[ℓ](\overline{ℚ}_{p})_{𝔪} is a faithful 𝕋_{𝔪}^{0}/ℓ-module. For this, it suffices to prove that J_{0}(p)[ℓ](\overline{ℚ}_{p}) is a faithful 𝕋^{0}/ℓmodule. Suppose that T ∈ 𝕋^{0} annihilates J_{0}(p)[ℓ](\overline{ℚ}_{p}). Then T is divisible by ℓ in End(J_{0}(p)). Since 𝕋^{0} is the full ring of endomorphisms of J_{0}(p) [15, prop. II.9.5], we see that in fact T is divisible by ℓ in 𝕋^{0}. This proves that J_{0}(p)[ℓ](\overline{ℚ}_{p}) is a faithful 𝕋^{0}/ℓ-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 𝕋_{𝔪}^{0} is Gorenstein if and only if J_{0}[𝔪] has rank two over 𝕋^{0}/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 is ordinary of residue characteristic two. Thus we need only prove the theorem in this case. We may furthermore assume that p > 2 (since X_{0}(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 the contravariant 𝔪-adic Tate module of J_{0}(p) is free of rank one over 𝕋_{𝔪}, and this is precisely the input required to deduce the theorem from [16]. (We remark that the reader 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.

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 Δ in N by proposition 1.1, it will suffice to show that the cokernel of θ has order Δ. 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 the cokernel of (3.10) has order Δ.

For this, it will suffice to show that for each prime number ℓ, the image of the morphism

(5.1)  X ⊗_{ℤ} ℤ_{ℓ} ⊗_{𝕋⊗_{ℤ}ℤ_{ℓ}} X ⊗_{ℤ} ℤ_{ℓ} ⟶ Hom_{ℤ_{ℓ}}(𝕋 ⊗_{ℤ} ℤ_{ℓ}, ℤ_{ℓ}),

obtained by tensoring the morphism (3.10) through by ℤ_{ℓ} over ℤ, and which is induced by the pairing (3.18), has cokernel of order equal to the maximal power of ℓ dividing Δ. Thus for the remainder of the proof we fix ℓ, and devote ourselves to proving this statement.

If we set A = ℤ_{ℓ}, B = 𝕋 ⊗_{ℤ} ℤ_{ℓ} and U = V = X ⊗_{ℤ} ℤ_{ℓ}, then the running hypotheses referred to in the statement of proposition 2.8 are satisfied. We will verify that for any maximal ideal of 𝕋 ⊗_{ℤ} ℤ_{ℓ}, either condition (i) or condition (ii) in the statement of proposition 2.8 is satisfied. For this, given any any maximal ideal 𝔪 of 𝕋 of residue characteristic ℓ, we must show either that the pairing (3.18) becomes perfect when restricted to X_{𝔪} × X_{𝔪} and that X_{𝔪}/ℓ is a faithful 𝕋_{𝔪}/ℓ-module, or that X_{𝔪} is a free 𝕋_{𝔪}-module of rank one.

If 𝔪 is of the first kind, then corollary 3.20 (ii) (and the isomorphism X_{𝔪}^{0} ⟶^{∼} X_{𝔪}) shows that the pairing (3.18) is perfect when restricted to X_{𝔪} × X_{𝔪}, while corollary 4.10 shows that X_{𝔪}^{0}/ℓ ⟶^{∼} X_{𝔪}/ℓ is a faithful 𝕋_{𝔪}/ℓ-module.

If 𝔪 is of the second kind, then X_{𝔪}^{Eis} ⟶^{∼} X_{𝔪} is isomorphic to ℤ_{ℓ}, and so is a free 𝕋_{𝔪} ⟶^{∼} 𝕋_{𝔪}^{Eis} ⟶^{∼} ℤ_{ℓ}-module of rank one, while if 𝔪 is Eisenstein, then theorem 4.2 shows that X_{𝔪} is a free 𝕋_{𝔪}-module of rank one.

Thus the hypotheses of proposition 2.8 are satisfied, and we deduce that the cokernel of (5.1) is isomorphic to the cokernel of the morphism X ⊗_{ℤ} ℤ_{ℓ} ⟶ Hom_{ℤ_{ℓ}}(X ⊗_{ℤ} ℤ_{ℓ}, ℤ_{ℓ}) induced by the pairing (3.18). Lemma 3.16 (i) shows that the cokernel of this morphism has order equal to the maximal power of ℓ dividing Δ. This completes the proof of theorem 0.3. □

Proof of theorem 0.4. We wish to prove that the morphism 𝕋 ⟶ End_{𝕋}(X) is an isomorphism. It suffices to prove this after tensoring this morphism through by ℤ_{ℓ}, for each prime number ℓ. Lemma 2.6 shows that it suffices to prove that X /ℓ is a faithful 𝕋/ℓ-module, for each prime number ℓ. Equivalently, it suffices to show that X_{𝔪}/ℓ is a faithful 𝕋_{𝔪}/ℓ-module for each maximal ideal 𝔪 of 𝕋 of residue characteristic ℓ. This was observed in the course of proving theorem 0.3. (When 𝔪 is of the second type or Eisenstein, note that X_{𝔪}/ℓ is even a free 𝕋_{𝔪}/ℓ-module of rank one, and so in particular is faithful.) □

Proof of theorem 0.5. Let ℓ denote the residue characteristic of 𝔪. Then 𝕋_{𝔪} is a free ℤ_{ℓ}-module of finite rank, and so is Gorenstein if and only Hom_{ℤ_{ℓ}}(𝕋_{𝔪}, ℤ_{ℓ}) is a free 𝕋_{𝔪}-module of rank one. Now proposition 1.3 (i) provides an isomorphism Hom_{ℤ_{ℓ}}(𝕋_{𝔪}, ℤ_{ℓ}) ⟶^{∼} N_{𝔪}, and so we see that 𝕋_{𝔪} is Gorenstein if and only if N_{𝔪} is free of rank one. Proposition 1.9 shows that N_{𝔪} is free of rank one if and only if M_{𝔪} is free of rank one. Putting these results together yields the equivalence of (i) and (iii).

The equivalence of (i)^{0} and (iii)^{0} is proved similarly (and is even more direct, since N^{0} = M^{0}), while the equivalence of (iii) and (iii)^{0} is simply a restatement of theorem 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. If 𝔪 is of the second type, then 𝕋_{𝔪} ⟶^{∼} 𝕋_{𝔪}^{Eis} ⟶^{∼} ℤ_{ℓ}, while 𝕋_{𝔪}^{0} = 0, and similarly X_{𝔪} ⟶^{∼} X_{𝔪}^{Eis} ⟶^{∼} ℤ_{ℓ}, while X_{𝔪}^{0} = 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 and the fact that a surjective morphism between 𝕋_{𝔪}-modules which are both free of rank one is necessarily an isomorphism. □

Proof of theorem 0.6. We begin by proving part (i). As usual, we let n denote the numerator of (p − 1)/12, expressed in lowest terms. We first note that M^{0}/I is of order n, by proposition 1.15. Thus in order to prove the theorem, we must show that the morphism

(5.2)  X^{0} ⊗_{𝕋^{0}} X^{0} ⟶ Hom_{ℤ}(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 the isomorphism of proposition 1.3 (ii).) For this it suffices to show, for each prime ℓ, that the morphism

X^{0} ⊗_{ℤ} ℤ_{ℓ} ⊗_{𝕋^{0}⊗_{ℤ}ℤ_{ℓ}} X^{0} ⊗_{ℤ} ℤ_{ℓ} ⟶ Hom_{ℤ_{ℓ}}(𝕋 ⊗_{ℤ} ℤ_{ℓ}, ℤ_{ℓ})

obtained by tensoring (5.2) through with ℤ_{ℓ} over ℤ, which is induced by the pairing (3.19), has cokernel annihilated by I and of order equal to the maximal power of ℓ dividing n.

The proof will be similar to that of theorem 0.3. Namely, we fix ℓ, and apply proposition 2.8, taking A = ℤ_{ℓ}, B = 𝕋^{0} ⊗_{ℤ} ℤ_{ℓ}, and U = V = X^{0} ⊗_{ℤ} ℤ_{ℓ}. That the 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 𝕋^{0} ⟶ End_{𝕋^{0}}(X^{0}) is an isomorphism, it suffices to prove that we obtain an isomorphism after tensoring this morphism through by ℤ_{ℓ}, for each prime number ℓ. Lemma 2.6 shows that it suffices to prove that X^{0}/ℓ is a faithful 𝕋^{0}/ℓ-module, for each prime number ℓ. Equivalently, it suffices to show that X_{𝔪}^{0}/ℓ is a faithful 𝕋_{𝔪}^{0}/ℓ-module for each maximal ideal of 𝕋^{0} of residue characteristic ℓ. That this is the case was observed in the course of verifying 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, the properties of the 𝕋-module X and the 𝕋^{0}-module X^{0} shown in the proofs of theorems 0.3 and 0.6, and the isomorphisms of proposition 1.3, once one notes that the pairings X × Hom_{ℤ}(X , ℤ) ⟶ ℤ and X^{0} × Hom_{ℤ}(X^{0}, ℤ) ⟶ ℤ are perfect by definition. □

6. Some remarks on the case when 𝕋_{𝔪} 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 ⟶^{∼} Hom_{A}(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 → End_{A}(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 Hom_{A}(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 of corollary 3.20 (ii) and the adjointness isomorphism (2.2) give rise to an isomorphism X_{𝔪}^{0} ⟶^{∼} Hom_{𝕋_{𝔪}^{0}}(X_{𝔪}^{0}, Hom_{ℤ_{ℓ}}(𝕋_{𝔪}^{0}, ℤ_{ℓ})). Since Hom_{ℤ_{ℓ}}(𝕋_{𝔪}^{0}, ℤ_{ℓ}) is a dualizing module for the one-dimensional local ring 𝕋_{𝔪}^{0}, this isomorphism together with theorem 0.4 shows that X_{𝔪}^{0} is a strong local theta-characteristic for 𝕋_{𝔪}^{0}; in particular, the ring 𝕋_{𝔪}^{0} does admit a strong local theta-characteristic. The preceding discussion shows that 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 J_{0}(p) when 𝕋_{𝔪} is non-Gorenstein.

Let us suppose that 𝔪 is an ordinary maximal ideal of 𝕋 of the first type, of residue characteristic ℓ ≠ p. Then the proof of [7, 12.10] and the discussion of [16] show that the 𝔪-adic Tate module T_{𝔪}J_{0}(p) of J_{0}(p) (that is, the localization at 𝔪 of the ℓ-adic Tate module of J_{0}(p)) sits in a short exact sequence of 𝕋_{𝔪}^{0} [Gal(\overline{ℚ}_{ℓ}/ℚ_{ℓ})]- modules

(6.2)  0 ⟶ T_{𝔪}J_{0}(p)^{0} ⟶ T_{𝔪}J_{0}(p) ⟶ T_{𝔪}J_{0}(p)^{e} ⟶ 0,

induced by the connected-étale short exact sequence of the ℓ-divisible group of the Néron model of J_{0}(p) over ℤ_{ℓ}. Furthermore, T_{𝔪}J_{0}(p)^{0} is isomorphic to 𝕋_{𝔪}^{0} as a 𝕋_{𝔪}^{0}-module, while T_{𝔪}J_{0}(p)^{e} is isomorphic to Hom_{ℤ_{ℓ}}(𝕋_{𝔪}^{0}, ℤ_{ℓ}).

Proposition 6.3. The short exact sequence (6.2) splits as a short exact sequence of 𝕋_{𝔪}^{0}-modules if and only if 𝕋_{𝔪}^{0} is Gorenstein.

Proof. If 𝕋_{𝔪}^{0} is Gorenstein, then Hom_{ℤ_{ℓ}}(𝕋_{𝔪}^{0}, ℤ_{ℓ}) is isomorphic to 𝕋_{𝔪}^{0}, and so (6.2) is an extension of free 𝕋_{𝔪}^{0}-modules, and hence splits. Thus to complete the proof of the proposition, it remains to show that (6.2) is non-split if 𝕋_{𝔪}^{0} is not Gorenstein.

Consider the short exact sequence

(6.4)  0 ⟶ J_{0}(p)[ℓ]_{𝔪}^{0} ⟶ J_{0}(p)[ℓ]_{𝔪} ⟶ J_{0}(p)[ℓ]_{𝔪}^{e} ⟶ 0

obtained by tensoring (6.2) through by ℤ/ℓ over ℤ_{ℓ}. We will show that (6.4) is non-split as a short exact sequence of 𝕋_{𝔪}^{0}/ℓ-modules.

Recall from theorem 4.6 that J_{0}(p)[ℓ]_{𝔪} is isomorphic to X_{𝔪}^{0}/ℓ ⊕ X_{𝔪}^{0}/ℓ as a 𝕋_{𝔪}^{0}/ℓmodule. Thus (6.4) is isomorphic (as a short exact sequence of 𝕋_{𝔪}^{0}/ℓ-modules) to a short exact sequence of the form

0 ⟶ 𝕋_{𝔪}^{0}/ℓ ⟶ X_{𝔪}^{0}/ℓ ⊕ X_{𝔪}^{0}/ℓ ⟶ Hom_{ℤ/ℓ}(𝕋_{𝔪}^{0}/ℓ, ℤ/ℓ) ⟶ 0.

It follows from the following lemma that there are no injective morphisms of 𝕋_{𝔪}^{0}/ℓmodules from Hom_{ℤ/ℓ}(𝕋_{𝔪}^{0}/ℓ, ℤ/ℓ) to X_{𝔪}^{0}/ℓ ⊕ X_{𝔪}^{0}/ℓ, 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 ℓ, then the surjection 𝕋_{𝔪}^{0}/ℓ ⟶ 𝕋_{𝔪}^{0}/𝔪 allows us to regard Hom_{ℤ/ℓ}(𝕋_{𝔪}^{0}/𝔪, ℤ/ℓ) as a submodule of Hom_{ℤ/ℓ}(𝕋_{𝔪}^{0}/ℓ, ℤ/ℓ).

Lemma 6.5. If 𝔪 is a maximal ideal of 𝕋 of residue characteristic ℓ for which 𝕋_{𝔪}^{0} is not Gorenstein, then any morphism from Hom_{ℤ/ℓ}(𝕋_{𝔪}^{0}/ℓ, ℤ/ℓ) to X_{𝔪}^{0}/ℓ contains Hom_{ℤ/ℓ}(𝕋^{0}/𝔪, ℤ/ℓ) in its kernel.

Proof. Note that since 𝕋_{𝔪}^{0} is assumed to be non-Gorenstein, theorem 1.14 shows that the maximal ideal 𝔪 is necessarily of the first type. We will write A = ℤ/ℓ, B = 𝕋_{𝔪}^{0}/ℓ, and U = X_{𝔪}^{0}/ℓ. 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 Hom_{A}(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 the conditions laid out in the preceding paragraph, then any morphism of B-modules ϕ : Hom_{A}(B, A) ⟶ U necessarily contains Hom_{A}(B/𝔫, A) in its kernel. Since Hom_{A}(B/𝔫, A) is a one-dimensional B/𝔫 vector space, if it is not in the kernel of ϕ then ϕ induces an injection Hom_{A}(B/𝔫, A) ⟶ U. Dualizing ϕ over A, we obtain a morphism ψ : Hom_{A}(U, A) ⟶ B which induces a surjection Hom_{A}(U, A) ⟶ B/𝔫. Thus ψ is itself a surjection, and since its source and target are of the same dimension over A, ψ is an isomorphism. But this is impossible, since Hom_{A}(U, A) ⟶^{∼} U is an A-self-dual B-module, while B is not, being non-Gorenstein by assumption. We conclude that Hom_{A}(B/𝔫, A) is in the kernel of ϕ, as claimed. □

Proposition 6.3 should be contrasted with the fact that T_{𝔪}J_{0}(p) sits in a short exact sequence of 𝕋_{𝔪}^{0} [Gal(\overline{ℚ}_{p}/ℚ_{p})]-modules

0 ⟶ Hom_{ℤ_{ℓ}}(X_{𝔪}^{0}, ℤ_{ℓ}(1)) ⟶ T_{𝔪}J_{0}(p) ⟶ X_{𝔪}^{0} ⟶ 0,

which is split as a short exact sequence of 𝕋_{𝔪}^{0}-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 𝕋_{𝔪}^{0} is non-Gorenstein.)

We close this section by describing concretely the 𝕋_{𝔪}^{0}/2-module X_{𝔪}^{0}/2 in the particular case when 𝔪 is a non-Gorenstein maximal ideal of residue characteristic 2 for which 𝕋_{𝔪}^{0}/2 has length four. Writing k = 𝕋_{𝔪}^{0}/𝔪, we note that 𝕋_{𝔪}^{0}/2 is then isomorphic to k [x, y, z]/(x^{2}, 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 𝕋_{𝔪}^{0} is non-Gorenstein and of rank four over ℤ_{2} [14], and so our description applies in particular to these cases.

Lemma 6.6. If 𝕋_{𝔪}^{0} is non-Gorenstein and 𝕋_{𝔪}^{0}/2 is of length four, then X_{𝔪}^{0}/2 [𝔪] has rank two over ℤ/2, and is equal to 𝔪X_{𝔪}^{0}/2.

Proof. Write d for the dimension of X_{𝔪}^{0}/2 [𝔪] over k. Applying lemma 4.4 with W = 𝕋_{𝔪}^{0}/𝔪 shows that X_{𝔪}^{0}/𝔪 also has dimension d. The short exact sequence

0 ⟶ 𝔪X_{𝔪}^{0}/2 ⟶ X_{𝔪}^{0}/2 ⟶ X_{𝔪}^{0}/𝔪 ⟶ 0

then shows that 𝔪X_{𝔪}^{0}/2 has dimension 4 − d. Since 𝔪^{2}𝕋_{𝔪}^{0}/2 = 0, we see that 𝔪X_{𝔪}^{0}/2 ⊂ X_{𝔪}^{0}/2 [𝔪], and so X_{𝔪}^{0}/2 [𝔪] has dimension at least 4 − d. Thus we see that d ≥ 2.

Since corollary 4.10 shows that X_{𝔪}^{0}/2 is a faithful 𝕋_{𝔪}^{0}/2-module, we see that the natural map 𝔪𝕋_{𝔪}^{0}/2 → End_{𝕋_{𝔪}^{0}/2}(X_{𝔪}^{0}/2) is injective. The image of this map lies in

Hom_{𝕋_{𝔪}^{0}/2}((X_{𝔪}^{0}/2)/(X_{𝔪}^{0}/2 [𝔪]), 𝔪X_{𝔪}^{0}/2).

Since 𝔪𝕋_{𝔪}^{0}/2 has dimension three, while Hom_{𝕋_{𝔪}^{0}/2}((X_{𝔪}^{0}/2)/(X_{𝔪}^{0}/2 [𝔪]), 𝔪X_{𝔪}^{0}/2) has dimension (4 − d)^{2}, we see that d ≤ 2. Combining this calculation with that of the preceding paragraph, we see that d = 4 − d = 2, and so deduce the lemma. □

The 𝕋_{𝔪}^{0}/2 action on X_{𝔪}^{0}/2 is determined by the action of the maximal ideal 𝔪𝕋_{𝔪}^{0}/2 on X_{𝔪}^{0}/2. This action in turn is determined by the composite

(6.7) 𝔪𝕋_{𝔪}^{0}/2 ⟶ Hom_{𝕋_{𝔪}^{0}/2}((X_{𝔪}^{0}/2)/(X_{𝔪}^{0}/2 [𝔪]), 𝔪X_{𝔪}^{0}/2)

= Hom_{k}(X_{𝔪}^{0}/𝔪, X_{𝔪}^{0}/2 [𝔪]) ⟶^{∼} X_{𝔪}^{0}/2 [𝔪] ⊗_{k} X_{𝔪}^{0}/2 [𝔪].

The final isomorphism holds because lemma 4.4 shows that the pairing X_{𝔪}^{0}/2 × X_{𝔪}^{0}/2 ⟶ ℤ/2 given by corollary 3.20 (ii) induces a duality between the k-vector spaces X_{𝔪}^{0}/2 [𝔪] and X_{𝔪}^{0}/𝔪. Since the pairing of corollary 3.20 (ii) is symmetric, the map (6.7) has image lying in the subspace Sym_{k}^{2}(X_{𝔪}^{0}/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 𝔪𝕋_{𝔪}^{0}/2 and Sym_{k}^{2}(X_{𝔪}^{0}/2 [𝔪]) both have dimension three over 𝕋^{0}/𝔪. Thus the 𝕋_{𝔪}^{0}/2-module structure on X_{𝔪}^{0}/2 is entirely determined by choosing an isomorphism of three-dimensional k-vector spaces 𝔪𝕋_{𝔪}^{0}/2 ⟶^{∼} Sym_{𝕋^{0}/𝔪}^{2}(X_{𝔪}^{0}/2 [𝔪]). This gives a complete description of the structure of X_{𝔪}^{0}/2 as a 𝕋_{𝔪}^{0}/2-module in the case when 𝕋_{𝔪}^{0}/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-prime level. 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.

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

f = ∑_{n=0}^{∞} a_{n}q^{n} on Γ_{0}(N) for which a_{n} ∈ ℤ (n ≥ 1), N (N)^{0} for the submodule of

N (N) consisting of cusp forms, and N (N)^{Eis} for the submodule of N (N) spanned by Eisenstein series.

We let 𝕋 denote the finite ℤ-algebra of Hecke operators acting on N (N), 𝕋^{0} denote the quotient of 𝕋 that acts faithfully on N (N)^{0}, and 𝕋^{Eis} denote the quotient of 𝕋 that acts faithfully on N (N)^{Eis}.

We have the following generalization of proposition 1.3 (whose proof can be found 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) ⟶^{∼} Hom_{ℤ}(𝕋, ℤ), defined by the pairing ⟨f, T ⟩ = a_{1}(f|T).

(ii) There is a natural isomorphism of 𝕋^{0}-modules N (N)^{0} ⟶^{∼} Hom_{ℤ}(𝕋^{0}, ℤ), defined by the same pairing as that of part (i).

Now suppose that N = Mp, where p is a prime which does not divide M. We let V_{p} denote the “degeneracy operator” f(q) ↦ f(q^{p}), which yields a morphism N (M) ⟶ N (Mp). The ℤ-submodule N (M) + N (M)|V_{p} of N (N) is stable under 𝕋, and we let 𝕋_{old} 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 𝕋_{old}. Then N (N)_{old} is the saturation of N (M) + N (M)|V_{p} in N (N), and so it contains N (M) + N (M)|V_{p} with finite index. Similarly, we write N (N)_{old}^{0} = N (N)^{0} ∩ N (N)_{old} and N (N)_{old}^{Eis} = N (N)^{Eis} ∩ N (N)_{old}. (Here the subscript “old” stands for “old at p”.)

The quotient of N (N)^{Eis} by N (N)_{old}^{Eis} is a free ℤ-module of rank one. Let E denote the weight two Eisenstein series on Γ_{0}(p), let Σ = {ℓ_{1}, ℓ_{2}, …, ℓ_{t}} denote the set of primes dividing M, and set

E_{new} = E′(q) − ∑_{ℓ∈Σ} E′(q^{ℓ}) + ∑_{{ℓα,ℓ_{β} ∈Σ \atop 1≤α<β≤t}} E′(q^{ℓ_{α}ℓ_{β}}) − ⋯ + (−1)^{t}E′(q^{ℓ_{1}···ℓ_{t}}).

(Here and below, the subscript “new” stands for “new at p”.) Then E_{new} is a normalized 𝕋-eigenform characterized by the conditions

E_{new}|T_{ℓ} = \{ \begin{matrix}(1 + ℓ)E_{new} if ℓ does not divide N, \\ ℓE_{new} if ℓ divides M, \\ E_{new} if ℓ = p, \end{matrix}

as ℓ ranges over all primes. One easily sees that E_{new} does not lie in N (N)_{old}^{Eis}, and so its image generates the quotient N (N)^{Eis}/N (N)_{old}^{Eis}.

We let 𝕋_{new}^{Eis} denote the quotient of 𝕋^{Eis} which acts faithfully on E_{new}. As a ring, 𝕋_{new}^{Eis} is isomorphic to ℤ. We let N (N)_{new}^{0} denote the maximal ℤ-submodule of N (N)^{0} orthogonal to N (N)_{old}^{0} under the Petersson inner product, and write 𝕋_{new}^{0} for the quotient of 𝕋^{0} which acts faithfully on N (N)_{new}^{0}. We also write 𝕋_{new} for the image of 𝕋 in 𝕋_{new}^{Eis} ⊕ 𝕋_{new}^{0}, and write N (N)_{new} for the maximal 𝕋-submodule of N (N) on which the 𝕋-action factors through 𝕋_{new} (so that N (N)_{new} is the saturation in N (N) of the submodule spanned by N (N)_{new}^{0} and E_{new}). We have 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 X_{0}(N) (regarded as a scheme over ℤ_{p}) 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 X_{0}(N) lie on different components; let us label the two components C_{0} and C_{∞}, according to which of these two specializations they contain. Each of these components is naturally isomorphic to X_{0}(M)_{/\overline{𝔽}_{p}} (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 of order M, then x is identified with the point on C_{0} obtained by taking the Frobenius twist of the pair (E, C). Let S = {x_{0}, …, x_{s}} denote the collection of singular points on X_{0}(N)_{/\overline{𝔽}_{p}}. By regarding each x_{i} as a point on C_{∞}, we may regard it as corresponding to a pair (E_{i}, C_{i}) 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 natural action of 𝕋 on X and X^{0}, which makes X a faithful 𝕋_{new}-module, and makes X^{0} a faithful 𝕋_{new}^{0}-module.

Let J_{0}(N) denote the Jacobian of X_{0}(N)_{/ℚ}. It is an abelian variety over ℚ, the special fibre of whose Néron model over ℤ_{p} has a semi-abelian connected component (that is, is the extension of an abelian variety by a torus). We let T_{0}(N) denote the maximal torus of this semi-abelian variety. There is a natural identification of X^{0} with the character lattice of T_{0}(N).

There is a monodromy pairing ⟨ , ⟩ : X × X ⟶ ℤ, defined by the formula

⟨x_{i}, x_{j}⟩ = \frac{1}{2}δ_{i,j}# Aut((E_{i}, C_{i})). This morphism is not 𝕋-bilinear if M > 1. However, it is easily modified to be so. As usual, let w_{M} denote the Atkin-Lehner involution on X_{0}(M), and define [ , ] : X × X ⟶ ℤ by the formula [x, y] = ⟨x, w_{M}y⟩. 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 X_{0}(N), we refer to sections 2 and 3 of [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 ⟶ Hom_{ℤ}(𝕋, ℤ) ⟶^{∼} N (N).

In fact, since X is a 𝕋_{new}-module, the image of θ lies in N (N)_{new}. Similarly to the case when M = 1, this morphism is closely related to the construction of certain theta series from quaternion algebras. However, the details are more complicated when M > 1.

If x_{i} and x_{j} are two elements of S, corresponding to the pairs (E_{i}, C_{i}) and (E_{j}, C_{j}) of supersingular elliptic curves and cyclic subgroups of order M, then we define L_{i,j} = Hom((E_{i}, C_{i}), (E_{j}, C_{j})) to be the ℤ-submodule of Hom(E_{i}, E_{j}) consisting of those homomorphisms ϕ : E_{i} ⟶ E_{j} for which ϕ(C_{i}) ⊂ C_{j}, equipped with the positive definite quadratic form given by taking the degree of an isogeny. The quadratic space L_{i,j} is rank four of level N = Mp. (In other words, if L_{i,j}^{∗} denotes the dual lattice to L_{i,j}, the cokernel of the injection L_{i,j} → L_{i,j}^{∗} induced by the quadratic form on L_{i,j} has exponent Mp.)

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 x_{i} and x_{j} as in the preceding paragraph, define the quadratic space L_{i,j}^{(d)} to be

L_{i,j}^{(d)} = Hom((E_{i}/C_{i}[d], C_{i}/C_{i}[d]), (E_{j}, C_{j}[M/d])),

regarded as the quadratic subspace of L_{i,j} consisting of homomorphisms containing C_{i}[d] in their kernel.

As above, let Σ = {ℓ_{1}, ⋯, ℓ_{t}} be the collection of prime factors of M.

Proposition 7.3. If x_{i} and x_{j} are any two elements of S, then

θ(x_{i} ⊗ w_{M}x_{j}) = \frac{1}{2}Θ(L_{i,j}) − ∑_{ℓ∈Q} \frac{1}{2}Θ(L_{i,j}^{(ℓ)})

+ ∑_{{ℓα,ℓ_{β} ∈Q \atop 1≤α<β≤t}} \frac{1}{2}Θ(L_{i,j}^{(ℓ_{α}ℓ_{β})}) − ⋯ + (−1)^{t} \frac{1}{2}Θ(L_{i,j}^{(ℓ_{1}···ℓ_{t})}).

Proof. If n is a positive integer, then

a_{n}(θ(x_{i} ⊗ w_{M}x_{j})) = a_{1}(θ(x_{i} ⊗ w_{M}x_{j})|T_{n}) = a_{1}(θ(T_{n}x_{i} ⊗ w_{M}x_{j}) = [T_{n}x_{i}, w_{M}x_{j}]

= ⟨T_{n}x_{i}, x_{j}⟩ = \frac{1}{2} the number of cyclic n-isogenies ϕ : E_{i} ⟶ E_{j} for which

ker(ϕ) ∩ C_{i} is trivial and ϕ(C_{i}) equals C_{j}

= \frac{1}{2} the number of ϕ ∈ L_{i,j} for which ker(ϕ) ∩ C_{i} is trivial.

If ϕ is an element of L_{i,j} for which ker(ϕ) ∩ C_{i} ≠ 0, then ker(ϕ) ⊃ C_{i}[q] for some prime q dividing M, and so ϕ ∈ L_{i,j}^{(q)}. 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 an algebro-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 vanishing constant term, and so in contrast to the situation of proposition 3.15, the question of 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 shows that 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 the question 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

induced by θ is an isomorphism of free 𝕋_{new} ⊗_{ℤ} ℚ-modules of rank one. Similarly, the morphism

X^{0} ⊗_{ℤ} ℚ ⊗_{𝕋^{0}⊗_{ℤ}ℚ} X^{0} ⊗_{ℤ} ℚ ⟶ N (N)_{new}^{0} ⊗_{ℤ} ℚ

induced by θ is an isomorphism of free 𝕋_{new}^{0} ⊗_{ℤ} ℚ-modules of rank one.

Proof. The ring 𝕋_{new} ⊗_{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 𝕋_{new} ⊗_{ℤ} ℚ-modules of the same dimension over ℚ as 𝕋_{new} ⊗_{ℤ} ℚ. Thus they are free 𝕋_{new} ⊗_{ℤ} ℚ-modules of rank 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 a similar fashion, or by localizing the isomorphism just proved at the direct summand 𝕋_{new}^{0} ⊗_{ℤ} ℚ of 𝕋_{new} ⊗_{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 together with the theta series Θ(L_{i,j}^{(d)}), as x_{i} and x_{j} range over the basis S of X, and d 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 natural morphism 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 like to 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 ℓ, then there is a semi-simple Galois representation ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) ⟶ GL_{2}(𝕋/𝔪) attached in a canonical fashion to 𝔪, characterized by the fact that for any prime q not dividing Nℓ, the representation ρ_{𝔪} is unramified at q, and the characteristic polynomial of ρ_{𝔪}(Frob_{q}) is equal to X^{2} − T_{q}X + q. Unfortunately we do not have control over the completion X_{𝔪} without making some restrictive hypotheses on both 𝔪 and ρ_{𝔪}, 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 ρ_{𝔪} is irreducible. Suppose in addition that

(i) the residue characteristic ℓ of 𝔪 is odd, and ℓ^{2} does not divide N; (ii) 𝔪 is not a prime of fusion between ℓ-old and ℓ-new forms in N (N)^{0} (this is

automatic if ℓ does not divide N, since then N (N)^{0} contains no ℓ-new forms);

(iii) if 𝔪 is a prime of fusion between p-old and p-new forms in N (N)^{0}, then at least one of the following holds: (a) ℓ does not divide p − 1, or (b) ρ_{𝔪}(Frob_{p}) does not act as a scalar.

Then the morphism X_{𝔪}⊗_{𝕋_{m}} X_{𝔪} ⟶ (N (N)_{new})_{𝔪} induced by θ is an isomorphism of free (𝕋_{new})_{𝔪}-modules of rank one.

Proof. We first note that since ρ_{𝔪} is irreducible, the natural maps 𝕋_{𝔪} ⟶ 𝕋_{𝔪}^{0}, (𝕋_{new})_{𝔪} ⟶ (𝕋_{new}^{0})_{𝔪}, and X_{𝔪}^{0} ⟶ X_{𝔪} are isomorphisms. We next observe that X_{𝔪} ⊗_{ℤ} ℚ is free of rank one over (𝕋_{new})_{𝔪} ⊗_{ℤ} ℚ, by proposition 7.4. Suppose we can prove that X /𝔪 is one-dimensional over 𝕋/𝔪 = 𝕋_{new}/𝔪; then since (𝕋_{new})_{𝔪} is reduced (by the results of [1]) we conclude that X_{𝔪} is a free (𝕋_{new})_{𝔪}-module of rank one. From [20, thm. 3.12/prop. 3.14/thm. 3.22] we deduce that the pairing X_{𝔪}×X_{𝔪} ⟶ ℤ_{ℓ} induced by [ , ] is perfect. Lemma 2.4 now shows that the morphism X_{𝔪} ⊗_{𝕋_{𝔪}} X_{𝔪} ⟶ (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 (𝕋_{new})_{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 ℓ is prime to N, this follows from [21, prop. 1] and hypothesis (iii) in the statement of the theorem. (As already noted, when ℓ is prime to N, hypothesis (ii) is superfluous.) Now suppose that ℓ exactly divides N. Hypothesis (ii) and the discussion of [22, p. 4] shows that ρ_{𝔪} appears with multiplicity one in J_{0}(N)[𝔪](\overline{ℚ}). If ρ_{𝔪} is not p-old, then the main theorem of [20] (which applies to our situation even though ℓ divides N, since we have the requisite statement regarding multiplicity one) shows that ρ_{𝔪} is not finite at p, and the same argument as used to prove [23, thm. 2.3] shows that X /𝔪 is one-dimensional over 𝕋/𝔪. If, on the other hand, ρ_{𝔪} is p-old, then by assumption (ii) ℓ ≠ p, and the proof of [21, prop. 1] then extends to again show that X /𝔪 is one-dimensional over 𝕋/𝔪 (the point as usual being that we have the requisite multiplicity one statement for J_{0}(N)[𝔪](\overline{ℚ})). □

Let S denote the following set of primes:

S = {ℓ | ℓ^{2} divides 4N}

∪ {ℓ | ℓ is a congruence prime between N (N)^{Eis} and N (N)^{0}}

∪ {ℓ | ℓ is a congruence prime between ℓ-old and ℓ-new forms in N (N)^{0}}

∪ {ℓ | ℓ 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)_{old}^{0}, together with the image of the map X^{0} ⊗_{𝕋^{0}} X^{0} ⟶ N (N)_{new}^{0} induced by θ, has index in N (N)^{0} divisible only by primes in S.

Proof. The set S contains all primes which occur as the residue characteristics of maximal ideals 𝔪 of 𝕋_{new} which do not satisfy the hypotheses of theorem 7.6, and so that theorem shows that the cokernel of the morphism X^{0}⊗_{𝕋^{0}} X^{0} ⟶ N (N)_{new}^{0} has order divisible only by primes in S. Furthermore, the index of N (N)_{new}^{0} + N (N)_{old}^{0} in N (N)^{0} is divisible precisely by the primes of congruence between N (N)_{new}^{0} and N (N)_{old}^{0}, and these are also contained in S. The theorem follows from these observations. □

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 that p = 11 and M = 3, so that N = 33. In this case there are no 11-old cusp forms, and so N (33)_{new}^{0} = N (33)^{0} and 𝕋_{new}^{0} = 𝕋^{0}. 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 J_{0}(33) is an elliptic curve, [22, cor. to thm. 2] shows that any maximal ideal of residue characteristic three satisfies multiplicity one, and so 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′ = {2, 5}. An explicit calculation shows in fact that the image of the map X^{0} ⊗_{𝕋^{0}} X^{0} ⟶ N (33)^{0} is of index ten in N (33)^{0}.

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 = Mp, with p not dividing M. We also assume that if M = 1 then p ≥ 5 (so that p is then prime to 2Δ, where as before Δ denotes the denominator of (p − 1)/12). Let Ω(cusps) denote the dualizing sheaf of X_{0}(N) over ℤ_{p}, twisted by the invertible sheaf corresponding to the relative divisor of cusps in X_{0}(N) over ℤ_{p}. The Hecke correspondences act naturally on H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) (see [15, §II.6] for a discussion in the case when M = 1) and this action serves to make H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) a 𝕋 ⊗_{ℤ} ℤ_{p}-module. In fact, one has the more precise statement given by the following result.

Proposition A.1. The ℤ_{p}-module H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) is naturally a 𝕋⊗_{ℤ}ℤ_{p}- submodule of N (N) ⊗_{ℤ} ℤ_{p} containing N (N)_{new} ⊗_{ℤ} ℤ_{p}.

Proof. We refer to [7, §8] for a discussion of the dualizing sheaf Ω of X_{1}(N) over ℤ_{p}[ζ] (where ζ is a primitive p^{th} root of unity) and its relation to weight-two cusp forms on Γ_{1}(N). (Note that our N is Gross’s Np.) In particular, [7, prop 8.4] shows that H^{0}(X_{1}(N)_{/ℤ_{p}[ζ]}, Ω) is a ℤ_{p}[ζ]-lattice inside the space of weight two cusp forms on Γ_{1}(N) defined over ℚ_{p}(ζ), which consists precisely of those cusp forms f such that both f and f|w_{ζ} have q-expansion coefficients lying in ℤ_{p}[ζ].

An identical argument to the one that proves this result applies in our situation to show that H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) is a ℤ_{p}-lattice inside N (N) ⊗_{ℤ_{p}} ℚ_{p}, consisting precisely of those modular forms for which both f and f|w_{p} have q-expansion coefficients lying in ℤ_{p}. (Here as usual w_{p} denotes the Atkin-Lehner involution at p on N (N)⊗_{ℤ}ℚ_{p} = N (Mp)⊗_{ℤ}ℚ_{p}.) Consequently, H^{0}(X_{0}(N), Ω(cusps)) is contained in N (N), and the action of the Hecke correspondences on H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) is obtained by restricting the 𝕋 ⊗_{ℤ} ℤ_{p}-action on N (N) ⊗_{ℤ} ℤ_{p}. We thus conclude that H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) is a 𝕋 ⊗_{ℤ} ℤ_{p}-submodule of N (N) ⊗_{ℤ} ℤ_{p}.

To show that H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps)) contains N (N)_{new} ⊗_{ℤ} ℤ_{p}, it suffices to show that if f ∈ N (N)_{new}⊗_{ℤ}ℤ_{p}, then both f and f|w_{p} have q-expansion coefficients lying in ℤ_{p}. By definition of N (N), all q-expansion coefficients of f lie in ℤ_{p}, except perhaps the zeroth; that is to say, the constant term of f. However this constant term vanishes if M > 1, while if M = 1 then we have assumed that p does not divide 2Δ, and so this constant term lies in ℤ_{p} in this case as well, by virtue of proposition 1.2. To see that all q-expansion coefficients of f|w_{p} also lie in ℤ_{p}, we note that f|w_{p} = −f|T_{p} if f ∈ N (N)_{new} ⊗_{ℤ} ℤ_{p} (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 = {x_{0}, …, x_{s}} denotes the set of singular points on the special fibre of X_{0}(N), that X denotes the free ℤ-module spanned by S, and that each point x_{i} ∈ S corresponds to a supersingular elliptic curve E_{i} over \overline{𝔽}_{p} together with

a cyclic subgroup C_{i} of E_{i} of order M. We write e_{i} = \frac{1}{2}# Aut((E_{i}, C_{i})).

Let W denote the Witt ring of \overline{𝔽}_{p}. After base-changing from ℤ to W, one finds (see the discussion of [20, p. 439]) that the formal completion of X_{0}(N) at the point x_{i} in its special fibre is isomorphic to the formal spectrum of

W [[x, y]]/(xy − p^{e_{i}}) = W [[x, \frac{p^{e_{i}}}{x}]].

Denote this ring by A_{i}. By writing the ring A_{i} 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 A_{i} (this annulus is the tube of the point x_{i} in the rigid analytic space underlying X_{0}(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 and do assume that the variables are labelled so that the differential dx/x yields this canonical orientation. The global sections of the dualizing sheaf of Spf A_{i} are a free

W [[x, \frac{p^{e_{i}}}{x}]]-module of rank one spanned by dx/x, which we denote Ω(A_{i}), and we have a map res_{i} : Ω(A_{i}) ⟶ W obtained by taking the residue of a section.

Proposition A.2. The morphism

(A.3)  H^{0}(X_{0}(N)_{/W} , Ω(cusps)) ⟶ X ⊗_{ℤ} W,

given by the formula ω ↦ ∑_{i} res_{i}(ω_{| Spf A_{i}})x_{i}, induces an isomorphism of 𝕋^{new}- modules

H^{0}(X_{0}(N)_{/W} , Ω(cusps))/(H^{0}(X_{0}(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 the taking of residues of this differential. The elements of H^{0}(X_{0}(N)_{/W} , Ω(cusps)) ∩ N (N)_{old} ⊗_{ℤ} W are rational multiples of pull-backs of differentials from X_{0}(M), a curve with good reduction modulo p, and so have trivial residues along the supersingular annuli of X_{0}(N). Thus (A.3) does induce a map

H^{0}(X_{0}(N)_{/W} , Ω(cusps))/(H^{0}(X_{0}(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 𝕋_{new}, and the source and target are both free W -modules of equal (finite) rank. Thus to show that this map is an isomorphism of 𝕋_{new}-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 the natural map H^{0}(X_{0}(N)_{/W} , Ω(cusps)) ⊗_{W} k ⟶ H^{0}(X_{0}(N)_{/k}, Ω(cusps)) is an isomorphism. Let C_{∞} and C_{0} denote the components of X_{0}(N)_{/k} containing the cusps ∞ and 0 respectively; they are both isomorphic to X_{0}(M)_{/k}. A section ω of H^{0}(X_{0}(N)_{/k}, Ω(cusps)) is determined by giving a pair of differentials of the third kind (ω_{∞}, ω_{0}), one on each of C_{∞} and C_{0}, regular away from the points x_{i} and the cusps, and such that the residues of each differential at each of the points x_{i} are negative to one another.

The reduction modulo p of the map (A.3) is given by

(ω_{∞}, ω_{0}) ↦ ∑_{i} res_{x_{i}}(ω_{∞})x_{i} ∈ X ⊗_{ℤ} k.

Since the set of cusps of both C_{∞} and C_{0} are non-empty, the residue theorem shows that this map is surjective, as required. □

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 argument used to prove this result makes it clear that it provides a kind of p-adic lift of the mod 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} ℤ_{p} is spanned by the theta series Θ(L_{i,j}). (A result stronger than Eichler’s theorem, but weaker than our theorem 0.3.)

Continuing to consider the case M = 1, note that if we consider the particular element E ∈ N (p) ⊗_{ℤ} W, then lemma 3.13 shows that the image of E under the isomorphism 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 X_{0}(p)_{/k} obtained by removing the cusp ∞ together with the supersingular points, and let X denote the open affinoid contained in X_{0}(p)_{/K} which is the preimage of C under specialization. Thus X is isomorphic to the complement of the supersingular residue class disks in the 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 punctures of X is equal to zero, and so we find that a_{0}(E) + w deg(x) = 0. Proposition 3.15 implies that w = −1/2Δ, and hence that the residue of E around the tube of x_{i} is equal to −1/2e_{i}. This result (or to be precise, its analogue when X_{0}(p) is replaced by the curve that classifies p-isogenies of elliptic curves equipped with full 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 the constant term of the Eisenstein series. As observed in the remarks following the proof of proposition 3.15, it would be interesting to have a purely algebro-geometric determination 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)_{new}^{0} ⊗_{ℤ} K ⟶^{∼} X^{0} ⊗_{ℤ} K, which is precisely the isomorphism obtained by taking k = 2 in [2, thm. 9.1] (up to the harmless replacement of Coleman’s Γ_{1}(M) ∩ Γ_{0}(p) level-structure by our Γ_{0}(Mp)- level structure). In some sense this is a p-adic analogue of the Riemann relations: it shows that the p-new differentials on X_{0}(N)_{/K} are uniquely determined by their periods 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 (𝕋_{old})_{𝔪} = 0, and either p ≥ 5 or the Galois representation ρ_{𝔪} is irreducible, then there is a natural isomorphism of 𝕋_{𝔪}-modules N (N)_{𝔪} ⟶^{∼} X_{𝔪}. Consequently both are free 𝕋_{𝔪}-modules of rank one, and so 𝕋_{𝔪} is Gorenstein.

Proof. Since (𝕋_{old})_{𝔪} = 0 by assumption, each of the morphisms 𝕋_{𝔪} ⟶ (𝕋_{new})_{𝔪}, (N (N)_{new})_{𝔪} ⟶ N (N)_{𝔪}, and

H^{0}(X_{0}(N)_{/W} , Ω(cusps))_{𝔪} ⟶

H^{0}(X_{0}(N), Ω(cusps))_{𝔪}/(H^{0}(X_{0}(N)_{/W} , Ω(cusps))_{𝔪} ∩ (N (N)_{old})_{𝔪} ⊗_{ℤ_{p}} W)

are isomorphisms. Thus propositions A.1 and A.2, combined with the flat basechange isomorphism H^{0}(X_{0}(N)_{/ℤ_{p}}, Ω(cusps))⊗_{ℤ_{p}} W ⟶^{∼} H^{0}(X_{0}(N)_{/W} , Ω(cusps)), yield an isomorphism of 𝕋_{𝔪}-modules

(A.5)  (N (N)_{new})_{𝔪} ⊗_{ℤ_{p}} W ⟶^{∼} X_{𝔪} ⊗_{ℤ_{p}} W.

Now the isomorphism of proposition 7.1 induces an isomorphism

(A.6) (N (N)_{new})_{𝔪} ⊗_{ℤ_{p}} W ⟶^{∼} Hom_{ℤ_{p}}((𝕋_{new})_{𝔪}, W)

⟶^{∼} Hom_{W} ((𝕋_{new})_{𝔪} ⊗_{ℤ_{p}} W, W).

Furthermore, the morphism (7.2) induces an isomorphism

(A.7)  X_{𝔪} ⟶^{∼} Hom_{ℤ_{p}}(X_{𝔪}, ℤ_{p}).

Indeed, if p ≥ 5 then this follows from the fact that the cokernel of the map (7.2) has exponent dividing twelve, while if ρ_{𝔪} is irreducible, we observed in the proof of theorem 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 ℤ_{p}, and combining it with the isomorphisms (A.5) and (A.6), we find that 𝕋_{𝔪} ⊗_{ℤ_{p}} W is a Gorenstein ring, and hence that the same is true of 𝕋_{𝔪}. □

We make one final remark in this appendix, again restricting ourselves to the case 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 free 𝕋 ⊗_{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,

⟨ω, ν⟩ = ∑_{i} e_{i} res_{x_{i}}(ω_{| Spf A_{i}}) res_{x_{i}}(ν_{| Spf A_{i}}).

It is interesting to compare this pairing with that of [3], which is defined precisely on those forms on Γ_{1}(p) whose traces to Γ_{0}(p) vanish.

The generator P is characterized in terms of this pairing by the condition ⟨ω, P ⟩ = a_{1}(ω) for any element ω of N (p) ⊗_{ℤ} W. Thus P could be thought of as a p-adic analogue of a Poincaré series.

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