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On maps between modular Jacobians and Jacobians of Shimura curves

By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new subvariety of J_0^D(pqN), whenever p and q are distinct primes not dividing D or N. However, this approach is entirely nonconstructive, and gives no information at all about any particular isogeny between the two varieties. In this paper, we determine the kernels of all possible maps from one variety to the other up to support on a small finite set of maximal ideals of the Hecke algebra. We also examine consequences of this computation for the geometry of Jacobians of Shimura curves. In particular, we obtain a"multiplicity one"result for certain Galois representations in the torsion of Jacobians of such curves.


Introduction
The Jacquet-Langlands correspondence asserts the existence of noncanonical isomorphisms between spaces of modular forms coming from different quaternion algebras over Q. More precisely, if D is a squarefree product of an even number of primes, p and q are distinct primes not dividing D, and B and B ′ are quaternion algebras over Q with discriminants D and Dpq, respectively, then the Jacquet-Langlands correspondence asserts the existence of an isomorphism: S Dpq 2 (Γ) ∼ = S D 2 (Γ 0 (pq) ∩ Γ) pq-new . Here S Dpq 2 (Γ) denotes the space of weight two modular forms associated to the quaternion algebra B ′ and the congruence subgroup Γ. Similarly, S D 2 (Γ 0 (pq) ∩ Γ) is the space of weight two modular forms associated to the quaternion algebra B and the congruence subgroup Γ 0 (pq) ∩ Γ. The subscript pq-new denotes the subspace of S D 2 (Γ 0 (pq) ∩ Γ) consisting of forms which are "new" for both p and q. The existence of such an isomorphism has consequences for the geometry of modular curves and Shimura curves. In particular, let X Dpq (Γ) denote the Shimura curve associated to the quaternion algebra B ′ and the congruence subgroup Γ 0 (pq) ∩ Γ, and define X D (Γ 0 (pq) ∩ Γ) similarly. Then we may consider their Jacobians J D (Γ 0 (pq)∩Γ) and J Dpq (Γ). The Jacquet-Langlands correspondence, together with Eichler-Shimura theory, tells us that, after tensoring with Q, the l-adic Tate module of J Dpq (Γ) is isomorphic to the corresponding Tate module of the pq-new subvariety of J D (Γ 0 (pq) ∩ Γ). It follows from [Ri3] or from Faltings' isogeny theorem [Fa], that J Dpq is isogenous to the pq-new subvariety of J D (Γ 0 (pq) ∩ Γ). D -new . Let T be the subalgebra of the endomorphism ring of some such J D generated by the Hecke operators T l for l prime, and the diamond bracket operators d . Repeated applications of the Jacquet-Langlands correspondence show that T is independent (up to canonical isomorphism) of the choice of J D used to define it, and therefore acts simultaneously on all the J D . Moreover, the abelian varieties J D are all T-equivariantly isogenous. We let A denote the category of abelian varieties with a T-action which are T-equivariantly isogenous to J, and develop a formalism which provides a concrete description of A in terms of T-modules.
In particular, we show that if one is willing to work "up to support on S", where S is the set of Eisenstein primes of T, that one can classify all abelian varieties in A in terms of isomorphism classes of T-modules. Specifically, to any abelian variety J isogenous to J 1 one can associate a certain T-module [J] which is free over Z and satisfies [J] ⊗ Q ∼ = T ⊗ Q. (Such modules were first introduced by Mazur [Ma], who called them "rank one" modules.) After formally inverting isogenies supported on S, and maps of T-modules whose cokernels are supported on S, this functor becomes an antiequivalence of categories (Theorem 4.11). Sections 3 and 4 are devoted to making this precise. Moreover, if J and J ′ are isogenous to J 1 , knowing the isomorphism classes of [J] and [J ′ ] as T-modules is often enough (Proposition 4.12) to determine the kernels of maps J → J ′ up to support on S. Thus computing [J] and [J ′ ] allows us to construct the isomorphism class of J ′ in terms of the isomorphism class of J. In particular one can express the Tate modules of J ′ at maximal ideals outside S in terms of the Tate modules of J and the modules [J] and [J ′ ].
Moreover, we obtain a relationship between the character groups X p (J) of the tori coming from the mod p reductions of J for p dividing N , and the module [J], for any J isogenous to J 1 (Proposition 4.14). In Section 5 we establish a key relationship between the character groups of J D as D varies. These two results allow us to compute the modules [J D ] explicitly. The key result we obtain is Theorem 6.1, in which we obtain [J D ] for D = N in terms of a certain module of maps between character groups.
Sections 6 and 7 are devoted to the proof of Theorem 6.1. In section 6, we introduce a condition, called controllability, associated to a prime p dividing N and a maximal ideal m of T. Controllability of m at p implies that the character group of a certain abelian variety isogenous to J D becomes free of rank one after completing at m (c.f. Lemma 6.5); in section 6 we exploit this fact to show that Theorem 6.1 holds locally at any maximal ideal m which is controllable at some prime dividing D.
It remains to handle those maximal ideals which are not controllable at any primes dividing D, which we do in Section 7. Although such ideals are not at all common, they are much more difficult to handle, and the arguments presented here are of a very technical sort. (The reader who is primarily interested in our results, or who is willing to take this extreme case of Theorem 6.1 on faith, is encouraged to read Section 8 first; once Theorem 6.1 is established we have no further need of the results of Section 7.) Our basic strategy for dealing with these maximal ideals is a level-raising argument. In particular, we fix such a maximal ideal m, and introduce two primes q 1 and q 2 to the level, in such a way that m is controllable at both q 1 and q 2 . We establish an analogue of Theorem 6.1 in this new setting, using controllability at q 1 and q 2 , and then show that it implies the desired result.
In Section 8 we explore the consequences of Theorem 6.1. In particular, we apply Theorem 6.1 along with Proposition 5.7 to obtain several expressions for [J D ] in terms of [J D ′ ] whenever D ′ divides D (Theorem 8.5, Theorem 8.7, and Corollary 8.14). In particular, if p is a divisor of D ′ , we find that the T-modules Hom(J D ′ , J D ) and Hom(X p (J D ), X p (J D ′ )) are isomorphic up to support on S, via the natural map which takes a morphism of varieties to the map it induces on character groups. Via Proposition 4.12, this allows us to determine the kernels of all morphisms J D → J D ′ , up to support on S. We give several different descriptions of these kernels (Corollaries 8.6,8.8,and 8.15) with varying degrees of explicitness. The latter two expressions, for instance, are in terms of character groups which can be computed algorithmically [Ko]. The upshot of this is that given the isomorphism class of J 1 , one can explicitly compute the isomorphism class of J D for any D, up to support on S.
This recipe for constructing J D from J 1 allows us to compute many invariants of J D in terms of J 1 . In particular one can compute the m-adic Tate modules of J D in this fashion, for any m outside S. In particular one obtains expressions for the dimension of J D [m] in this manner. In this fashion, we obtain a bound on the dimension of J D [m] in terms of the number of primes dividing D at which m is not controllable, a significant strengthening of an earlier result of L. Yang [Ya]. An important special case of this bound is when m is controllable at every prime and J 1 [m] has dimension two; in this case J D [m] has dimension two as well-that is, we have "multiplicity one" for J D at m.
Hecke algebra. If U is a divisor of T , we let J D (Γ) U -old (resp. J D (Γ) U -new ) denote the connected components of the intersections of the varieties J D (Γ) p-old (resp. J D (Γ) p-new ) for p dividing U .
We denote by T D (Γ) the subalgebra of End(J D (Γ)) generated by the Hecke operators T l for l prime to N , and the diamond bracket operators d for d in (Z/N Z) × . (When l divides N we take the T l to be defined as in [Ri1].) The algebra T D (Γ) is a finite, flat Z-algebra. If Γ = Γ 0 (T ) ∩ Γ ′ as above, and U divides T, then T D (Γ) acts on J D (Γ) U -new (resp. J D (Γ) U -old ) through a finite quotient which is also flat over Z. We denote this quotient by T D (Γ) U -new (resp. T D (Γ) U -old ).
If J is an abelian variety with a T D (Γ)-action, then this action allows us to speak of the U -new subvarieties and quotients of J as well. We take the U -new subvariety of J to be the connected component of the subvariety of J annihilated by the kernel I of the map T D (Γ) → T D (Γ) U -new , and the U -new quotient of J to be J/IJ. The U -old subvariety and quotient of J are constructed in a similar manner. It is straightforward to check that this construction agrees with the previous definition of J D (Γ) U -new and J D (Γ) U -new .
We have the following relationship between the subvarieties and quotients defined above and duality of abelian varieties: Proof. Consider the exact sequence: Taking duals, we obtain an exact sequence: (Here the action of T on J ∨ is induced by duality of abelian varieties.) It thus suffices to show that we can identify A ∨ with IJ ∨ , where I is the kernel of the map T D (Γ) → T D (Γ) U -new . Since I annihilates (J U -new ) ∨ , we must have IJ ∨ ⊂ A. Counting ranks of abelian varieties, we find that the two must be equal. ✷ Fix a particular D and N , a congruence subgroup Γ ′ of level N , and two distinct primes p and q which do not divide DN . Let Γ = Γ ′ ∩ Γ 0 (pq). The tangent spaces S D 2 (Γ) pq-new and S Dpq 2 (Γ ′ ) of J D (Γ) pq-new and J Dpq (Γ ′ ) over Q, respectively, are modules for T D (Γ) pq-new and T Dpq (Γ ′ ).
Theorem 2.3 (Jacquet-Langlands, Ribet) There is a natural isomorphism: T D (Γ) pq-new → T Dpq (Γ ′ ) which sends an operator T l in one algebra to the corresponding operator T l in the other (and similarly for d .) Moreover, the T Dpq (Γ ′ )-modules S D 2 (Γ) pq-new and S Dpq Thus J and J ′ are T-equivariantly isogenous.
✷ We finish this section with some notation and conventions. We will often be considering modules over a finite, flat Z-algebra T. If M is a T-module, and m is a maximal ideal of T, then M m denotes the completion of M at m. If M is free as a Z-module, then M * denotes the Z-dual Hom Z (M, Z), with the T-module structure induced from M . If M is free as a Z l -module for some l, on the other hand, then by abuse of notation we write M * for Hom Z l (M, Z l ). The advantage of this notation is that (M m ) * is naturally isomorphic to (M * ) m for any maximal ideal m of T.
The T-modules we are most interested in are all torsion free over Z. Unfortunately, the tensor product of two T-modules without Z-torsion often has Z-torsion, which we wish to ignore. Therefore we adopt the convention that if M and M ′ are torsion free T-modules or T m -modules, then M ⊗ M ′ denotes their tensor product (over T or T m ) modulo Z-torsion. It can be verified that this operation satisfies the universal property of the tensor product in the category of T-modules without Z-torsion. Note that if I and I ′ are ideals of T, then I ⊗ I ′ is isomorphic to II ′ under this convention, as the natural map sending x ⊗ x ′ to xx ′ is a surjection by construction, and the modules in question have the same rank over Z. Thus the kernel is finite, and hence trivial since under our convention I ⊗ I ′ is torsion free.
We will find the following observation about these two operations to be useful in what follows: Lemma 2.5 Let M and N be finitely generated, torsion free T-modules. We have natural isomorphisms: Proof.
1. We have M * ∼ = Hom T (T, M * ). Taking Z-duals, we find that the latter is naturally isomorphic to Hom T (M, T * ). (1), (M ⊗N ) * is canonically isomorphic to Hom T (M ⊗N, T * ), which in turn is canonically isomorphic to Hom T (M, Hom T (N, T * )), and the result follows.

By
3. This follows from (2) by replacing N with N * and dualizing. ✷

Morphisms "up to support on S"
In what follows, we will often find it convenient to work in categories of abelian varieties in which certain morphisms have been formally inverted. In particular, we will fix a finite flat Z-algebra T and an abelian variety J /Q on which T acts faithfully.
Definition 3.1 Let A be the category whose objects consist of abelian varieties J ′ with an action of T, such that there exists a T-equivariant isogeny φ : J → J ′ defined over Q, and whose morphisms are T-equivariant maps of abelian varieties defined over Q.
We are interested in the behavior of maps in this category up to support on a certain finite set S of maximal ideals of T. In order to exclude phenomena that happen solely at maximal ideals inside S, we wish to formally invert morphisms in A which are isomorphisms at all m outside S.
This motivates us to consider the category A S of "abelian varieties up to S-isogeny", whose objects are the same as the objects of A, but whose morphisms include formal inverses for those isogenies with kernel supported on S. In order to make this precise, we invoke the following lemma: Lemma 3.2 Let S be a finite set of maximal ideals of T. Then there is an element σ S of T such that for any maximal ideal m of T, σ S ∈ m if and only if m ∈ S. In particular, U = Spec T − S is affine, and equal to Spec T[ 1 σS ].
Proof. It suffices to show this when T is reduced, as if σ red S is an element of T red (the reduced quotient of T) with the desired property, then any lift of σ red S to T will have the desired property. Suppose T is reduced, and letT be its normalization. ThenT is a product of maximal orders in number fields, and so its Picard group is finite. In particular, for each maximal ideal m of T, we can find an element σ m ofT such thatσ m lies in mT but does not lie in m ′T for any m ′ different from m. Lemma 3.3 For n sufficiently large,σ n m lies in T. Proof. Let N be the index of T inT. It suffices to check this statement locally at each maximal ideal of T containing N . Sinceσ m lies in mT, for some n σ n m lies in NT m and therefore lies in T m . Fix a maximal ideal m ′ of T distinct from m but containing N . Thenσ m lies in T × m ′ . Let n be the order of (T m ′ /N T m ′ ) × . Then σ n m ∈ 1 + N T m ′ ⊂ T m ′ , as required.
✷ We now resume the proof of Lemma 3.2. Let σ m =σ n m . Then σ m lies in m, but does not lie in m ′ for any m ′ distinct from m. Taking σ S equal to the product of the σ m gives us a σ S with the desired properties. ✷ Let T S = T[ 1 σS ]. We define A S to be the category whose objects are the objects of A, but whose morphisms are given by Hom is defined by composition either on the left or the right; since morphisms in A are T-equivariant it does not matter which.) This has the desired effect of ignoring behavior supported on S, as any morphism whose kernel is supported on S factors through a sufficiently high power of σ S , and hence is an isomorphism in A S . We will make use of this construction in the next section.
Observe that since σ S is a unit in T m for any m outside S, any morphism , for n sufficiently large to make σ n S a morphism in A. It will be convenient to speak of the "kernel" of a map in A S , even though strictly speaking this notion is not well defined. It is possible to make it precise "up to support on S", however; for any morphism f : A 1 → A 2 in A S , we can fix an n such that σ n S f is a morphism in A, and take ker S f = ker σ n S f . This is independent of n up to a finite group supported on S. Definition 3.4 Let B be a subvariety of A 1 which is stable under T and closed under addition and inverses, and f : A 1 → A 2 a morphism in A S . Then ker S f = B "up to support on S" if for some n, σ n S f is a morphism in A, ker σ n S f contains B, and the group (ker σ n S f )/B is finite and supported on S. (Equivalently,

The isogeny class of an abelian variety with T-action
We now begin the program described in the introduction, in which we study the category of abelian varieties isogenous to J D (Γ) for some D and Γ. Our techniques apply in considerably more generality, however, so rather than working explicitly with J D (Γ), we work in the following setting: let J be an abelian variety, defined over Q, and suppose that J comes equipped with the action of a finite, flat Z-algebra T, in which the action of any element σ of T on J is defined over Q.
If m is a maximal ideal of T of residue characteristic l, let T m J denote the m-adic contravariant Tate module of J at m; that is, the inverse limit of J[m k ] ∨ over k, where (−) ∨ denotes Cartier duality. Thus T m J is naturally isomorphic to Hom(J[m ∞ ], µ l ∞ ). If α ∈ T m J and x ∈ J[m ∞ ], we denote by α(x) the root of unity obtained by evaluating the element of Hom(J[m ∞ ], µ l ∞ ) which corresponds to α at the element x. For a morphism f : A 1 → A 2 of abelian varieties with T-action, we let f m denote the induced map T m A 2 → T m A 1 .
The action of T on J induces an action of T⊗ Q on the singular cohomology H 1 (J(C), Q) of J; we assume that this makes H 1 (J(C), Q) into a free T ⊗ Q-module, of rank two. In particular, this means that the action of T on J is faithful.
Since there are natural isomorphisms H 1 (J(C), Z) m ∼ = T m J, T m J ⊗ Q is free of rank two as a T m ⊗ Qmodule. The action of G Q on T m J ⊗ Q gives us a two-dimensional representation ρ m of G Q over T m ⊗ Q. We impose the further hypothesis that Tr ρ m (g) and det ρ m (g) (a priori elements of T m ⊗ Q), lie in T m for all m and g. We further assume that the subalgebra of T m generated by the elements Tr ρ m (g) and det ρ m (g) as g varies is reduced for all m.
Finally, we assume that there exists a T-equivariant map ψ : J → J ∨ , defined over a finite extension K of Q, and a character χ of Gal(K/Q), taking values in T × , such that ψ has finite kernel, and for all σ ∈ Gal(K/Q), we have σψ = χ(σ)ψσ.

Remark 4.1
The motivation for these assumptions comes from the example in which J = J D (Γ) for some discriminant D and congruence subgroup Γ of level N , and T is the Hecke algebra T D (Γ). In this case it can be shown (see for instance Carayol [Ca]) that H 1 (J(C), Q) is free of rank two over T ⊗ Q. Moreover, for a prime l not dividing N D, Tr ρ m (Frob l ) is equal to T l ∈ T, and det ρ m (Frob l ) = l l . The subalgebra of T generated by l and T l for l prime to N D is indeed reduced, as it is well-known that there exists a basis of simultaneous eigenforms for all such T l . Since the elements Frob l for l prime to N are dense in G Q , and T m is complete, the subalgebra of T m generated by the traces and determinants of elements of G Q is generated by the aforementioned l and T l , and is therefore reduced. Finally, the map ψ described above is constructed in Lemma 5.5. It follows that the hypotheses of this section are satisfied in this case.
The existence of the map ψ described above has strong consequences for the structure of T. In particular, ψ induces an isomorphism T m J ⊗ Q → (T m J) * ⊗ Q. Since the former is free of rank two over T m ⊗ Q, and the latter is isomorphic to (T * m ⊗ Q) 2 , it follows that T * m ⊗ Q is free of rank one over T m ⊗ Q. In particular, T ⊗ Q is a zero-dimensional Gorenstein ring.
Following the analysis of the Tate modules of modular Jacobians in [Ca], we fix a maximal ideal m of T, and let T ′ m denote the subalgebra of T m generated by the traces and determinants of ρ m (g) for g ∈ G Q . By our hypothesis, this is a reduced subalgebra of T m . We can thus consider the normalizationT ′ m of T ′ m ; this will be a product of discrete valuation rings O i , with corresponding fields of fractions K i . Let k i denote the residue field of O i ; then we may consider the mod m reduction ρ m,i : G Q → GL 2 (k i ) of ρ m,i . For g ∈ G, Tr ρ m,i (g) is equal to the reduction of Tr ρ m (g) modulo m, and is therefore independent of i. Thus the semisimplification of ρ m,i is independent of i.
With this definition, we make one final assumption on J, namely, that for all but finitely many m, the representations ρ m,i are absolutely irreducible. For such m, ρ m,i is independent of i and we denote it simply by ρ m .
Remark 4.2 It should be pointed out here that in the case in which J is a modular Jacobian, the ρ m constructed above is the Cartier dual of the representation usually called ρ m in the literature. (This is because the latter is defined in terms of the m-torsion of J, whereas the representations we consider are defined in terms of T m J instead.) Fix a finite set S of maximal ideals of T, containing all m for which one (and hence all) of the ρ m,i is not absolutely irreducible. The following result is a straightforward generalization of [Ca], Theorem 3: Proof. Consider the action of G Q onV ′ m = (T ′ m ) 2 in which G Q acts on each factor (O i ) 2 of (T ′ m ) 2 by ρ m,i . This gives a representation of G Q in GL 2 (T ′ m ) whose character agrees with ρ m . By [Ca], Theorem 2, this representation is realizable over T ′ m ; i.e., there is a G Q -module V ′ m free of rank two over T ′ m such that the character of the G Q -action agrees with that of ρ m . Then V m = V ′ m ⊗ T ′ m T m is free of rank two over T m , and V m ⊗ Q is equivalent to ρ m by [Ca], Theorem 1.
✷ Our goal is to study T-equivariant isogenies of J which are defined over Q, up to support on S. The first step in this study is to understand the structure of T m J for m outside S. Lemma 4.4 For all but finitely many maximal ideals m of T, T m J is free of rank two over T m .
Proof. Since we have assumed that H 1 (J(C), Q) is free of rank two over T⊗ Q, we can choose two elements of H 1 (J(C), Z) that generate H 1 (J(C), Q) over T ⊗ Q. These two elements give us a map T 2 → H 1 (J(C), Z), with finite cokernel. Localizing this map at any m on which the cokernel has no support gives the desired isomorphism T 2 m ∼ = T m J. ✷ We will see (Lemma 4.6, below) that the Tate modules at such m are particularly well-behaved. It will thus be convenient for us to replace J with an isogenous abelian variety, which we call J min , for which T m J min is free of rank two over T m for all m outside S. The following lemma allows us to do so. Lemma 4.5 There exists an abelian variety J min , with a T-action satisfying the above hypotheses, such that J and J min are T-equivariantly isogenous (over Q), and T m J min is free of rank two over T m for all m outside S.
Proof. Let S ′ be the set of m outside S at which T m J is not free of rank two; then S ′ is finite. For each m in S, fix an isomorphism V m ⊗ Q ∼ = T m J ⊗ Q. Multiplying this isomorphism by a sufficiently large n ∈ Z, we may assume that it maps V m into T m J, with finite index. Let W m ⊂ J[m ∞ ] be the finite subset of the m-divisible group annihilated by the image of V m in T m J, under the natural pairing If we take J min to be the quotient of J by the sum of the W m , we find that the map T m J min → T m J identifies T m J min with the image of V m for all m in S ′ , and is an isomorphism for m outside S ′ ; in either case T m J min is free of rank two over T m . ✷ We fix, for the remainder of this section, a J min as above. Proof. We use induction on the length of a maximal G Q -stable filtration of T m J min /M . If this length is zero, then I is the unit ideal and the result is clear. In general, let M = M n ⊂ M n−1 ⊂ · · · ⊂ M 0 = T m J min be a maximal G Q -stable chain of submodules. By induction we may assume that M n−1 = I ′ T m J min for some I ′ .
Consider mM n−1 + M . This is G Q -stable, contains M , and is contained in M n−1 . It is thus equal to either M or M n−1 . Suppose it were equal to M n−1 . Then M/mM n−1 would equal M n−1 /mM n−1 , which is impossible since then M would contain a set of generators for M n−1 by Nakayama's lemma. Thus mM n−1 + M = M , so M contains mM n−1 .
Let V ρ m be a two-dimensional (T/m) vector space on which G Q acts via ρ m . The module M n−1 /mM n−1 is G Q -equivariantly isomorphic to (I ′ /mI ′ ) ⊗ T/m V ρ m , where G Q acts trivially on I ′ /mI ′ . Let V be the image of M in M n−1 /mM n−1 . Since V is G Q -invariant, and V ρ m is irreducible, V is given byV ⊗ V ρ m for some subspaceV of I ′ /mI ′ . Let I be the preimage ofV in I ′ . Then IT m J = M , since both contain mM n−1 and map to V modulo mM n−1 .
✷ This result has important consequences for T-equivariant maps from J min . In particular, for any abelian varieties A 1 , A 2 which are T-equivariantly isogenous to J over Q, let Hom(A 1 , A 2 ) (resp. End(A 1 )) denote the T-module of T-equivariant maps: A 1 → A 2 (resp. T-equivariant endomorphisms of A 1 ) which are defined over Q. Then: Proof. Fix m outside S; we show that the natural map is an isomorphism. We have an isomorphism End(J min ) m ∼ = End G (T m J min ), by [Fa], Theorem 4. End G (T m J min ) is clearly faithful over T m , and finitely generated. It thus suffices to show that End G (T m J min ) ⊗ T/m is 1-dimensional. Observe that End G (J min [m]) is 1-dimensional, by Schur's lemma and the (absolute) irreducibility of ρ m . We construct an injection: the existence of which immediately implies the desired result. The exact sequence induces an exact sequence: and the latter is isomorphic to End G (J min [m]) via Cartier duality. Now Hom G (T m J min , mT m ) clearly contains m End G (T m J min ), and it suffices to show that they are equal. Let φ ∈ Hom G (T m J min , mT m ); by Lemma 4.6, the image of φ is isomorphic to IT m J min for some I ⊂ m. Since IT m J min ∼ = T m J min and T m J min is free over T m , I is principal; say generated by σ. Thus φ/σ is a well defined endomorphism of T m J min . Since σ ∈ m, φ lies in m End G (T m J min ) as required. ✷ Proof. The map φ induces an exact sequence: In particular the image of T m A is a T-stable submodule of T m J min . By Lemma 4.6, we can find an ideal m , the result follows. ✷ Note that the above result implies that the isomorphism class of I m does not depend on the isogeny φ, but only on the variety A. In fact, this is true globally, as well as locally, as long as one works "up to support on S". To see this, fix a φ and I as above. Then for σ ∈ I, ker φ ⊂ J min [I] ⊂ J min [σ]. Thus the map σ : J min → J min factors through φ; i.e. σ = ϕφ for a unique map ϕ : A → J min . Let f to be the map which associates to each σ the corresponding map ϕ. Proof. The map f must be injective, as if σ induced the zero map A → J min , then the action of T on J min could not be faithful. It remains to show that f m is surjective for m outside S.
Let ϕ be an element of Hom(A, J min ). Then the composition ϕφ lies in Hom(J min , J min ). By Corollary 4.7, the inclusion T m → Hom(J min , J min ) m is an isomorphism. Thus we can choose an element σ of T m corresponding to ϕφ; since J min [I] m is contained in the kernel of ϕφ by Corollary 4.8, σ lies in I m . Moreover, f m (σ) = ϕ by construction. ✷ Corollary 4.10 If A is T-equivariantly isogenous to J, and m lies outside S, then the natural map Proof. Since T m J min is free over T m , T m J min ⊗ Hom(A, J min ) m is torsion free. It thus suffices to show that the natural map is surjective, as the modules in question have the same Z-rank.
Fix an isogeny φ : J min → A. Then φ identifies T m A with IT m J min , where I = Ann(ker φ). Let x ∈ T m A; then φ identifies x with an element y of IT m J min ; we write y as a sum of elements of the form σ i y i with σ i ∈ I and y i ∈ T m J min . Since σ i ∈ I, the map σ i : J min → J min factors through φ, i.e., we have maps ϕ i such that σ i = ϕ i φ. Then σ i y i = ϕ i (φ(y i )), so φ(y i ) ⊗ ϕ i maps to σ i y i under the natural map. Thus each of the σ i y i are in the image, so y is in the image as well. The second statement follows immediately since T m J min is free of rank two over T m . ✷ Thus to any A which is T-equivariantly isogenous to J we can associate the module Hom(A, J min ). This module is torsion free as a Z-module, and Hom(A, J min ) ⊗ Q is free of rank one as a T ⊗ Q-module by Proposition 4.9. Following the previous section, we let A be the category whose objects consist of abelian varieties A such that there exists a T-equivariant isogeny φ : J → A defined over Q, and whose morphisms are T-equivariant maps of abelian varieties defined over Q. Then A → Hom(A, J min ) is a well defined functor A → M, where M denotes the category of finitely generated T-modules M such that M is torsion free over Z and M ⊗ Q is free of rank one over T ⊗ Q.
Let T S be the T-algebra defined in the previous section, and consider the category A S defined therein. The functor defined above induces a functor A S → M S , where M S is the category of "rank one" T Smodules, i.e., the category of finitely generated T S -modules that are torsion free over Z and become free of rank one over T S ⊗ Q. Proof. We must show that the functor is full, faithful, and dense. Given an M ∈ M, we can find an injection M → T. Let I be its image. Then by Proposition 4.9, we have a map I → Hom(J min /J min [I], J min ) whose cokernel is supported on S. In particular, Hom(J min /J min [I], J min ) is isomorphic to M in M S . Thus the functor is dense.
It remains to show that the functor is fully faithful. Since Hom AS (A 1 , A 2 ) and Hom MS ( is an isomorphism for any m outside S and any A 1 , A 2 ∈ A. We have a commutative diagram: , in which the bottom horizontal map is induced by the isomorphisms [ Note that each map appearing in this diagram is injective, as the modules in question have no Z l -torsion and each map becomes an isomorphism after tensoring with Q. Moreover, the left-hand vertical map and the bottom horizontal map are isomorphisms. Thus the righthand vertical map is surjective, and hence an isomorphism. It follows that the upper horizontal map is an isomorphism as well. ✷ In light of this result, we will primarily be interested in objects and morphisms in A S and M S for the remainder of this chapter. Rather than clutter our notation by writing "⊗ T T S " repeatedly, we adopt the following convention: if M and N are objects of M, an S-morphism: The formalism we have introduced gives information about isogenies in A, up to support on S, but it does so in terms of our choice of J min , which was somewhat arbitrary. Our next goal is to obtain a result that does not involve this choice. Let us assume that we have objects A 1 and A 2 in A; then we can study the maps A 1 → A 2 by studying maps from [ ] by the same locally free T-module.) In this case we can characterise maps from A 1 to A 2 in terms of M . Recall that we have adopted the convention that if M and M ′ are objects of M, then M ⊗ M ′ is taken to mean the usual tensor product modulo Z-torsion (and thus also lies in M).
If we have an S-morphism h : By Theorem 4.11, we can considerĥ to be a map M → Hom(A 2 , A 1 ). This gives us a map g : Proposition 4.12 The map g defined above is an S-isomorphism. In particular, g induces an isomorphism Proof. We construct an inverse to g in M S . We have a sequence of natural S-morphisms: where the first and last morphisms come from Theorem 4.11, and the second sends a map f to the map f ⊗ 1. It is straightforward to check that this provides an inverse to g. Now fix a φ ∈ Hom AS (A 1 , A 2 ), and an m outside S, and let I φ be the corresponding ideal defined above.
is surjective. (The latter map can be obtained from the former by tensoring with T m J min and exploiting the isomorphism Thus we can write any α ∈ T m A 2 as a sum of terms of the form for some m i in M m and β i in T m A 1 . But by our construction of g, we have Conversely, observe that by construction, every element of I φ factors through φ, since any element of I φ is in the image of g(φ) and the definition of g(φ) involved composition with φ. Thus the image of φ m contains I φ T m A 1 .
It follows that φ m T m A 2 = I φ T m A 1 for all m outside S. Since the former is the annihilator of ker φ : ✷ The behavior of certain invariants of J under isogeny can be studied using this formalism. In particular, let p be a prime, and suppose that J has purely toric reduction at p. Then the special fiber of the Nerón model of J at p is an extension of a finite group scheme by a torus T , and we can consider the character group X p (J) = Hom Fp (T, G m ) of T . More generally, any object A of A will also have purely toric reduction at p, and we can consider its character group X p (A).

Lemma 4.13
The character group X p (A) is a "rank one" T-module.
Proof. By construction, X p (A) has finite rank over Z, and is torsion free as a Z-module. It thus suffices to show that X p (A) ⊗ Q is free of rank one as a T ⊗ Q-module. Let p be a maximal ideal of T ⊗ Q. It is then enough to show that X p (A) p is free of rank one over T ⊗ Q. Since is generated (as a (T ⊗ Q)/p-vector space) by a single element σ. Since T acts faithfully on J, it acts faithfully on A as well, and hence on X p (A). In particular there is an x in X p (A) p not killed by σ. Let I be the annihilator of x in T ⊗ Q. Then I is killed by some power of p, so I[p] is nontrivial unless I is the zero ideal. But if I[p] were nontrivial, then I would contain σ, which does not annihilate x by assumption. Thus I is the zero ideal, and X p (A) p contains a free (T ⊗ Q) p -submodule for all p.
Counting Q-ranks, we find that X p (A) p is free of rank one over (T ⊗ Q) p for all p, since the rank of X p (A) is equal to the dimension of A, and hence to the Z-rank of T.
✷ If A 1 and A 2 are two objects in A, and φ : ). By abuse of notation, we denote this map by φ as well. (It will always be clear from the context whether we mean a morphism of abelian varieties or the map it induces on the character groups.) (1)), by the results of [SGA], chapter 5, this gives us a functorial surjection: These surjections fit into the commutative diagram: in which the horizontal maps are the natural ones, the right hand vertical map is the surjection defined above, and the left hand vertical map is the surjection defined above tensored with the identity. By Corollary 4.10, the upper horizontal map is surjective; since both vertical maps are surjections the lower horizontal map must be surjective as well. ✷ We will also have use for the following result, which is an immediate consequence: Corollary 4.15 Let m be a maximal ideal outside S, and suppose that X p (J min ) m is free of rank one over T m . Then for any Proof. By the proposition, and the fact that X p (A) m is free of rank one over T m , we have a natural isomorphism: and T m J min is free of rank two, the result follows. ✷ Finally, we wish to interpret duality of abelian varieties in terms of the formalism we have constructed. This presents a minor difficulty; namely, the dual of an abelian variety J in A need not lie in A. In fact, the abelian variety J ∨ will be a "twist" of some abelian variety in A by the character χ we fixed at the beginning of this section, in a sense we will make precise below.
Let A χ denote the category of abelian varieties which are T-equivariantly isogenous to J ∨ , and whose morphisms are T-equivariant and defined over Q. Duality of abelian varieties yields a functor A → A ∨ from A to A χ .
Consider also the category A 1,χ , whose objects are triples (A, Lemma 4.16 The functors A 1,χ → A and A 1,χ → A χ are equivalences of categories. Proof. Both functors are clearly fully faithful; it suffices to show that they are dense. Fix an A ∈ A; then it is enough to show that we have an A ′ ∈ A χ and a map ϕ : A → A ′ with the properties described above. Fix an isogeny φ : J → A. Multiplying by a sufficiently large integer n, we may assume that ker φ contains the kernel of the map ψ : J → J ∨ . Let A ′ = J ∨ /ψ(ker φ). The natural map J → A ′ induced by ψ has the same kernel as φ, and thus descends to an isomorphism ϕ from A to A ′ . Since ψ is defined over K, so is ϕ, and we have σϕ = χ(σ)ϕσ since the corresponding relation holds for ψ. It follows that (A, A ′ , ϕ) is an object of A 1,χ mapping to A.
The proof for the functor A 1,χ → A χ is similar. ✷ The upshot is that A and A χ are equivalent categories. In particular the above equivalences induce a functor A → A χ from A to A χ and the inverse functor A ′ → A ′ χ −1 , and these functors are equivalences of categories. (These functors can be thought of as "twisting" an abelian variety A by the character χ or its inverse. In particular A and A χ become isomorphic over the splitting field of χ, via an isomorphism ϕ which satisfies σϕ = χ(σ)ϕσ for all σ in Gal(K/Q).) This allows us to defined a natural duality operation on A, by taking This operation interacts with our equivalence of categories A S → M S as follows: Proof. The Weil pairing induces an isomorphism, (1)).
Since T m (J min ) ∨ and T m (J min ) † are isomorphic as T m -modules (but not Galois modules), T m (J min ) † is isomorphic to Hom(T m J min , Z l (1)). As T m J min is free of rank two over T m , it follows that (T m (J min ) † ) * is free of rank two over T m . By Corollary 4.10, [(J min ) † ] * is free of rank one over T m . Now for any A ∈ A, we have The character group of the pq-new subvariety We now return to the situation considered in the introduction. Fix a squarefree integer N , with an even number of prime divisors, and let Γ be a congruence subgroup of level M for some integer prime to N . For R a divisor of N , and D a divisor of R which is divisible by an even number of primes, we let J D,R denote the abelian variety J D (Γ 0 (N/D) ∩ Γ) R D -new . Let T R denote the subalgebra of End(J 1,R ) generated by the Hecke operators T l for l prime, (when l divides M N we take T l to be defined as in [Ri1]), and the diamond bracket operators, and let A R denote the category of abelian varieties defined over Q which are T R -equivariantly isogenous to J 1,R . Then for each D as above, J D,R is an object of A R . Since we will be primarily interested in A N , we let J D = J D,N for any divisor D of N which is divisible by an even number of primes. Similarly, we take A = A N , T = T N , and so forth. Let S R be the set of maximal ideals of T R which are either Eisenstein, or of residue characteristic 2 or 3, and take S = S N .
As mentioned in Remark 4.1, the hypotheses (and hence the results) of the previous section hold for A R . We fix, once and for all, a J min 1 inside A 1 such that T m J min 1 is free of rank two over T 1 m for all non-Eisenstein m. If we then take J min R = (J min 1 ) R-new , then T m J min R is free of rank two over T R for all non-Eisenstein maximal ideals m of T R . We can thus consider the functors J → [J] R = Hom(J, J min R ) for J ∈ A R , and use them to study isogenies between the abelian varieties J D,R for various D. By abuse of notation, we often denote [J] R simply by [J] if it is clear from the context which category J belongs to. J 1,R (and hence every variety in A R ) has purely toric reduction at any prime p dividing R. Thus we can consider the character groups of J D,R at such primes. We then have relationships between these character groups and the module [J D,R ], given by Proposition 4.14. Thus, if we can compute the character groups X p (J D,R ), we can hope to obtain information about maps between varieties of this form from this data. We carry out this program in the remainder of this paper.
Let R ′ be an integer dividing N and divisible by R. By [CS], Theorem 8.2, the injection J D,R ′ → J D,R induces an injection on the maximal tori of the special fibers at p, and hence a surjection X p (J D,R ) → X p (J D,R ′ ). Moreover, both X p (J D,R ) and X p (J D,R ′ ) are free Z-modules, and there are isomorphisms: This actually suffices to determine the isomorphism class of X p (J D,R ′ ) in terms of X p (J D,R ).
More generally, let A be any finite flat Z-algebra, and suppose that A ⊗ Q is Gorenstein. (This is easily seen to hold for A = T R by results of the previous section.) Let A ′ be a quotient of A which is flat over Z. Let X be a finitely generated R-module such that X ⊗ Q is free of rank one over A ⊗ Q, and Y be a quotient of X that is a faithful A ′ -module, and flat over Z.
Lemma 5.1 Suppose X and Y are as above, and let L be the kernel of the map X → Y . For an ideal I of A, let I ⊥ be the annihilator of I in A. (This notation is due to Emerton [Em].) Let K be the kernel of the map A → A ′ . Then L = X[K ⊥ ], and we have Y ∼ = X ⊗ A A ′ (recall our convention, in which we take the tensor product to be the usual tensor product modulo Z-torsion).
Proof. We begin by showing that K has finite index in K ⊥⊥ . (Note that K ⊥⊥ trivially contains K.) Sincê A = A ⊗ Q is a zero-dimensional Gorenstein ring, we fix an isomorphismÂ ∼ = Hom Q (Â, Q). For any ideal I ofÂ, this isomorphism identifies I ⊥ with Hom Q (Â/I, Q). In particular dim Q I ⊥ = dim QÂ − dim Q I, and so dim Q I ⊥⊥ = dim Q I. Since I is contained in I ⊥⊥ , the two must be equal. It follows that K is contained in K ⊥⊥ with finite index, as required.
We have an exact sequence: In particular L contains KX with finite index. Hence Y ∼ = X/L is isomorphic to X/KX, modulo Z-torsion in X/KX. The latter is precisely X ⊗ A A ′ . ✷ We record the following useful corollary before we proceed: Corollary 5.2 Let X, Y , A, A ′ , and K be as above. Then Y * ∼ = X * [K]. In particular X * [K] ∼ = (X ⊗ A A ′ ) * for any X and K as above.
Taking duals gives us the sequence: The result now follows by the previous lemma. ✷ The upshot of Lemma 5.1 is the following corollary: Corollary 5.3 Let R and R ′ be divisors of N , and suppose R divides R ′ . For any p dividing R, we have We now fix two distinct primes p and q dividing N . Let D be a divisor of N , divisible by an even number of primes but not divisible by p or q. The previous result, together with the following result of Ribet, establishes a relationship between the character groups X p (J D,Dpq ) and X q (J Dpq,Dpq ): Theorem 5.4 There is a T D -equivariant exact sequence: Proof. When D = 1, this is [Ri1], Theorem 4.1. The same argument extends to the case where D > 1 using Buzzard's analogue of the Deligne-Rapoport theorem [Bu1], except that Buzzard's result assumes that the level structure is contained in Γ 1 (r) for some integer r > 4. An argument that removes this restriction can be found in the appendix. ✷ Dualizing, and applying Lemma 5.1, we find that This is almost what we need; it remains to compare X p (J D,D ) * with X p (J D,D ). We do this via the monodromy pairing on X p (J D,D ). This is a bilinear pairing: for which the adjoint of T l acting on J D,D is T ∨ l acting on (J D,D ) ∨ .
Lemma 5.5 Let Γ ′ = Γ 0 (N/D) ∩ Γ, and R = N M/D be the level of Γ ′ . There exists a Hecke-equivariant isomorphism ψ : Proof. Let w R be the Atkin-Lehner involution on Γ ′ = Γ 0 (N/D)∩Γ; that is, the involution of J D,D = J D (Γ ′ ) corresponding to the double coset Γ ′ 0 −t R 0 Γ ′ . The discussion on page 193 of [Sh], together with [Sh], Proposition 3.54, shows that the Rosati involution induced by the canonical polarization θ of J D (Γ ′ ) sends an element T l of T to w R T l w R , for any l prime to D. (Strictly speaking, Shimura only covers the case in which D = 1 and l is arbitrary, but one can just as easily work locally at l for any l prime to D and his argument carries over without change.) For l dividing D, one can view X D (Γ ′ ) as a moduli space of abelian surfaces with quaternionic multiplication by the quaternion algebra B of discriminant D, with associated level structure. (See [Bu1] for details.) From this perspective, T l is induced by the map on X D (Γ ′ ) which sends an abelian surface A to A/A[l], where l is the unique two-sided maximal ideal of index l in a maximal order of B D . Since this map is an isomorphism, the Rosati involution sends T l to its inverse. But it is easy to see from the moduli definition of T l that T 2 l = l . Thus the Rosati involution sends T l to l −1 T l . Moreover, by [Sh], Proposition 3.55, we have w R T l w R = l −1 T l on J 1 (Γ 0 (D) ∩ Γ ′ ), and hence on J D (Γ) as well by Jacquet-Langlands.
Finally, it is easy to verify that both the Rosati involution and conjugation by w R send d to d −1 for any d. (For the former, simply note that d is induced by an automorphism of X D (Γ ′ ); the latter is part of [Sh], Proposition 3.55). Since the elements T l and d as l and d vary generate T, we find that the Rosati involution is simply conjugation by w R . It is then easy to check that ψ = w R θ is Hecke-equivariant. The second statement (and the fact that ψ is defined over Q(µ R )), follow from the identity σw R = χ(σ)w R σ. ✷ The map ψ defined above induces a duality operation A → A † , by results of the previous section. It also defines an isomorphism J D,D ∼ = (J D,D ) ∨ over Q(µ R ). Since the character groups at p of (J D,D ) ∨ are the same whether we consider the varieties over Q or over Q(µ R ), the map ψ induces an isomorphism X p (J D,D ) ∨ ∼ = X p (J D,D ) (as T D -modules but not Galois modules). Since ψ is T D -equivariant, in the sense that T ∨ l ψ = ψT l , this identification allows us to view the monodromy pairing Proof. Since p is prime to D, the methods of [Ri4], which cover the discriminant 1 case, carry over to the case of arbitrary discriminant once one invokes Buzzard's analogue of the Deligne-Rapoport Theorem ( [Bu1]), and works with abelian surfaces with quaternionic multiplication rather than elliptic curves. As before, see the appendix for an extension of Buzzard's result to the case in which the level structure does not contain Γ 1 (r) for some r > 4. ✷ With this in hand, we can now show: Proof. Corollary 5.3 identifies X p (J D,Dpq ) with X p (J D,D ) ⊗ T D T Dpq . On the other hand, the Z-dual of the exact sequence of Theorem 5.4 identifies X q (J Dpq,Dpq ) * with the module X p (J D,D ) * ⊗ T D T Dpq by Lemma 5.1. Since X p (J D,D ) ∼ = S D X p (J D,D ) * by the above discussion, it follows that X p (J D,Dpq ) ∼ = S Dpq X q (J Dpq,Dpq ) * . ✷ We will also have need of a compatibility property which the above isomorphisms possess. Let p and p ′ be two distinct primes dividing D, take D ′ = D pp ′ , and choose distinct primes q 1 and q 2 dividing N but not D. Let q = q 1 q 2 .
Lemma 5.8 Let f : J D ′ q,D ′ q → J D ′ ,D ′ and g : J Dq,Dq → J D,D be maps such that there is a commutative diagram: in which the the horizontal maps are those of Theorem 5.4. Let f denote the restriction of f to the pp ′ -new subvariety J D ′ q,Dq of J D ′ q,D ′ q . Then we have a commutative diagram: in which the horizontal maps are the isomorphisms constructed in Proposition 5.7.
Proof. Consider the diagram: Here the left-hand square is that appearing in the hypothesis of the lemma. The middle square commutes because f † and f are adjoints under the monodromy pairing. The right-hand square is the Z-dual of the diagram: which clearly commutes since f is the restriction of f to the pp ′ -new subvariety, and the horizontal maps are just inclusions of the pp ′ -new subvarieties into the corresponding variety. The isomorphisms: are obtained as the compositions of the two leftmost horizontal map with the "inverse" of the rightmost horizontal map (bearing in mind that the image of the rightmost horizontal map is the same as the image of the composition of the two rightmost), in the corresponding row of the above diagram. The commutativity of the desired square thus follows immediately. ✷ We will also need the following result about character groups in section 7.
Proposition 5.9 Let q 1 and q 2 be two distinct primes dividing N but not D. Let Y q1 = X q1 (J Dq1q2,Dq1q2 ) and let X q2 = X q2 (J D (Γ 0 (N/Dq 1 ) ∩ Γ)) 2 . Then there is a natural S D -isomorphism of T D -modules: (Here Y q1 is considered as a submodule of Y * q1 via the monodromy pairing.) Proof. In [Ri1], Ribet constructs an exact sequence: where K and C are supported on S D , in the case D = 1 and Γ = Γ 0 (N ). As with Theorem 5.4, his arguments carry over to our setting once one invokes the analogue of the Deligne-Rapoport theorem found in the appendix. ✷

Induced maps on character groups
In light of the character group computations of the previous section, we might hope to study morphisms in A N by studying the maps they induce on character groups. The main tool we will use to do this is the following result, whose proof will occupy this section as well as the next. which sends a map J min D → J D,D to the induced map on character groups at p is an S D -isomorphism.
We will prove this theorem by constructing isomorphisms: for each m outside S D . Although the isomorphisms we construct in this way are in some sense natural, showing any sort of compatibility between any two of them seems difficult. The following lemma provides us a way around this difficulty:  (Hom(M, N )). Then there would be a maximal ideal m 1 of End(Hom(M, N )) containing σ. Since End(Hom(M, N )) is contained in End(N ) with finite index, there is a maximal idealm 1 of End(N ) containing m 1 . In particular,m 1 would contain σ, and so σ would not be a unit in End(N ). This is a contradiction, so σ is a unit in End(Hom(M, N )), as required.
✷ The usefulness of this lemma can be seen in the following corollary: Corollary 6.3 Let J ∈ A D , p be a prime dividing R, m a maximal ideal of T R outside S D , and suppose that [J] m ∼ = Hom(X p (J min D ), X p (J)) m . Then the natural map [J] m → Hom(X p (J min D ), X p (J)) m is an isomorphism.
Proof. We have a natural isomorphism: By Lemma 6.2, this induces a map [J] m → Hom(X p (J min D ), X p (J)) m , which is easily seen to be the natural map [J] m → Hom(X p (J min D ), X p (J)) m . Since the two modules are isomorphic, this map is an isomorphism by Lemma 6.2. ✷ In particular, in order to prove Theorem 6.1, it suffices to construct, for each m outside S D , an arbitrary isomorphism [J D,D ] m → Hom(X p (J min D ), X p (J D,D )) m . We begin by doing this for maximal ideals m with a certain property.
Definition 6.4 Suppose D divides N , and is divisible by an even number of primes, and let m be a maximal ideal of T D such that the representation ρ m is absolutely irreducible. Then m is controllable at p (for some p dividing N ) if one of the following conditions holds: 1. ρ m is not finite at p, or 2. ρ m is unramified at p, p = l, and ρ m (Frob p ) is not a scalar, or 3. p = l, and l = 2, or 4. p = l = 2, and the restriction of ρ m to a decomposition group at 2 is not contained in the scalar matrices.
The point of this definition is the following lemma: Lemma 6.5 Suppose that m is controllable at p. Then X p (J min D ) m is free of rank one over T D m .
Proof. This is essentially Mazur's principle. By Nakayama's lemma, and the fact that X p (J min D ) m is faithful over T D m , it suffices to show that the quotient X p (J min D )/mX p (J min D ) has dimension one. We will prove the lemma in this form.
Let is not finite at p, and so X p (J min D )/mX p (J min D ) has dimension one. We may thus assume that ρ m , and hence J min D [m], is finite at p. Suppose further that X p (J min D )/mX p (J min D ) has dimension two. Then we are in the following situation: J min D [m] extends to a finite flat group scheme W over Z p , and the special fiber of W over F p is the multiplicative group scheme Hom(X p (J min D )/mX p (J min D ), µ l ). In particular, the Cartier dual W ∨ of W is anétale group scheme over Z p . Since Gal(Q/Q) acts on the general fiber of W ∨ by ρ m , it follows that ρ m is unramified at p. In particular, the image of a decomposition group at p under ρ m is determined by the action of Frob p on the special fiber of W ∨ = X p /mX p .
Suppose first that p = l. Then Frob p acts on X p (J min D )/mX p (J min D ) via the Atkin-Lehner operator w −1 p , as X p (J min D ) can be identified with a subspace of X p (J 1,D ), on which this is well-known. Since w p acts as the scalar −T p on this space, it follows that ρ m (Frob p ) is a scalar. In particular, m cannot be controllable at p in this case.
Suppose finally that p = l. Since the determinant of ρ m is the mod l cyclotomic character times an even character of finite order, ρ m is ramified at p unless p = l = 2. This contradicts the fact that W ∨ isétale. If p = l = 2, then we find as before that the action of Frob p on W ∨ = X p /mX p is by scalars, and hence m again cannot be controllable at p. ✷ When this criterion is satisfied for some prime dividing D, we may easily construct the desired isomorphism: Corollary 6.6 Let p divide D, and suppose m is not Eisenstein and controllable at at least one prime dividing D. Then [J D,D ] m is isomorphic to Hom(X p (J min D ), X p (J D,D )) m for any m outside S D .
Proof. If m is controllable at p, then X p (J min D ) m is free of rank one over T m , and The result follows immediately.
If m is not controllable at p, there is another prime q dividing D at which m is controllable. Then we have a sequence of isomorphisms: This proves the first statement; the second statement follows by Corollary 6.3. ✷ It remains to deal with those maximal ideals m that are not controllable at any prime dividing D. We do so in the next section, by "raising the level". Specifically, we introduce two auxiliary primes q 1 and q 2 , prime to N M , such that m lifts to a maximal ideal ofm of T D (Γ 0 (q 1 q 2 N/D) ∩ Γ) which is new at q 1 and q 2 and controllable at these primes. Thenm descends to a maximal ideal of T Dq1q2 (Γ 0 (N/D) ∩ Γ) (becausẽ m is new) and an analogue of Corollary 6.6 holds for the abelian variety J Dq1q2 (Γ 0 (N/D) ∩ Γ) atm, asm is controllable at q 1 . We then exploit a geometric relationship between this abelian variety and J D,D to prove the desired result for J D,D .

Raising the level
The key to the level raising argument is the following result: Lemma 7.1 Let D be the product of an even number of distinct primes, and Γ a congruence subgroup of level M prime to D. Let m be a non-Eisenstein maximal ideal of the Hecke algebra T D (Γ), of residue characteristic l > 3. Then there exists a prime q, not dividing 2N Dl, such that: is not a scalar, and 2. There is a maximal idealm of T D (Γ 0 (q) ∩ Γ) q-new such that the representations ρ m and ρm are isomorphic.
Proof. Let χ be the character such that det ρ m = χω, where ω is the cyclotomic character. Let σ be the representation ρ m × χ. By theČebotarev density theorem, we may choose q such that σ(Frob q ) is conjugate to σ(c), where c denotes complex conjugation. Arguments of Diamond [Di] show that ρ m (Frob q ) then has order two, trace zero and determinant −1. In this setting (1) is immediate; (2) follows from [DT], Theorem A. ✷ Suppose we fix a q andm as above. Thenm is q-old; that is, it descends to an ideal of the algebra T D (Γ 0 (q) ∩ Γ) q-old . We now investigate the structure of this ring. The two degeneracy maps This isogeny is compatible with a certain action of the Hecke operators on J D (Γ) 2 , in which the diamond bracket operators and the operators T n for n prime to q act diagonally, and the action of T q is given by the matrix τ q q q −1 0 .
(Here τ q denotes the endomorphism of J D (Γ) 2 induced by the qth Hecke operator acting diagonally.) A proof may be found in [Di]. It follows that T D (Γ 0 (q) ∩ Γ) q-old is isomorphic to R[T q ]/(T 2 q − τ q T q + q q ), where R is the subalgebra of T D (Γ) generated by the diamond bracket operators and the Hecke operators T n for n prime to q. By the lemma on p. 491 of [Wi], R is all of T D (Γ). Moreover, the Eichler-Shimura relation implies thatm ∩ R = m.
Proposition 7.2 Let q andm be chosen as above. Then Proof. To make the notation more concise, letJ = J D (Γ 0 (q) ∩ Γ),J new =J q-new , andJ old =J q-old . Let J = J D (Γ). As described above, the two degeneracy maps J →J induce a Hecke-equivariant isogeny (J 2 ) →J old . By [DT], Theorem 2, the kernel of this isogeny is Eisenstein; in particular it is supported away fromm.
LetJ old be the q-old quotient ofJ (and thus the dual ofJ old ), and consider the sequence of maps: where the map ψ is given by Lemma 5.5, and the map J old → J 2 is the dual of the isogeny J 2 →J old .
Calculations of Ribet [Ri1] show that the composition of these maps is given by the matrix q + 1 q T q T q q + 1 , and is in particular equal to T 2 q − q times an automorphism of J 2 . Let ψ old :J old →J old be the map i † ψi, above. The kernel of ψ old is equal to the intersection ofJ old with the kernel of the mapJ →J old ; the latter is preciselyJ new . Since the intersection ofJ old withJ new is finite, andJ old andJ old have the same dimension, ψ old is surjective. Since the map J 2 →J old is an isogeny, and T 2 q − q : J 2 → J 2 factors as it follows that the map T 2 q − q is a surjective endomorphism of J 2 . We immediately see that the map T 2 q − q :J old →J old is surjective as well. Finally, observe that since the kernel of the map J 2 →J old is supported away fromm, it follows that them-primary part of ker ψ old is equal to them-primary part of J old [T 2 q − q ]. The operator T q acts onJ new as −w q , where w q is the Atkin-Lehner operator. Since w 2 q = q , it follows that T 2 q − q annihilatesJ new . The endomorphism ofJ induced by this map thus factors throughJ old ; its image therefore lies in the q-old subvarietyJ old . Since (T 2 q − q )J old is equal toJ old , this image must be all ofJ old . Now consider the diagram: The Snake lemma then yields an exact sequence: The final map in this sequence is simply the inclusionJ old ∩J new →J new , and hence injective, so this yields a short exact sequence: Takingm ∞ -torsion then yields The discussion above showed that J old [T 2 q − q ]m lies in the kernel of ψ old , and hence lies in q of the special fiber of the Nerón model ofJ at q. SinceJ has semistable reduction at q, it fits into an exact sequence 0 → Hom(X q (J), G m ) →J 0 q →J 2 → 0.
Since the natural isogeny J 2 →J old has kernel supported away from m, it suffices to show that J 2 [m] has dimension 2n.
The new subvarietyJ new has purely toric reduction at q, so the special fiber of its Nerón model at q is an extension of the torus Hom(X q (J new ), G m ) by the finite group Φ q (J new ). As ρ m is unramified at q, and J new [m] has dimension 2n, (J new ) q [m] has dimension 2n. We have an exact sequence: Moreover, sincem is controllable at q (ρ m (Frob q ) is not a scalar), it follows by Lemma 6.5 and Corollary 4.15 that TmJ new ∼ = X q (J new ) 2 m . In particular, X q (J new )/mX q (J new ) has dimension n. The above exact sequence then shows that Φ q (J new )[m] has dimension at least n. (We will see later that it must have dimension exactly n.) The inclusionJ new →J induces an injection on the special fibers (J new ) q andJ q of the corresponding Nerón models, and in particular an isomorphism X q (J new ) → X q (J). Since the group of connected components Φ q (J) is Eisenstein (c.f. Proposition 5.6), Φ q (J new )[m] maps into the connected component ofJ q . Since the map on special fibers is injective, the image of Φ q (J new )[m] inJ q has trivial intersection with the torus, and hence maps injectively into J 2 .
We thus obtain an injection of . The latter is a direct sum of two dimensional subspaces V i on which Frobenius acts via ρ m (Frob q ). Since Frob q acts as a scalar on Φ q (J new ), the image of Φ q (J new ) in each V i has dimension at most one. As Φ q (J new )[m] has dimension at least n, it follows that J 2 [m] has dimension at least 2n. But dim J 2 [m] = dimJ old [m], and the latter is less than dimJ[m] which is 2n. Thus dimJ old [m] = dimJ[m], as required.
✷ We now return to the setting of the previous section; we have fixed a D dividing N which is the product of an even number of primes, a maximal ideal m, and a prime p dividing D, and we wish to show that [J D,D ] m ∼ = Hom(X p (J min ), X p (J D,D )) m . Let p ′ be a prime dividing D other than p, and let D ′ = D pp ′ . Now we choose two primes q 1 and q 2 as in Lemma 7.1. The choice of these primes determines several abelian varieties. In particular, the abelian varieties J D ′ (Γ 0 (q 1 q 2 N/D ′ ) ∩ Γ) and J D (Γ 0 (q 1 q 2 N/D) ∩ Γ) arise from J D ′ ,D ′ and J D,D by raising the level at q 1 and q 2 . We denote them byJ D ′ ,D ′ andJ D,D , respectively. We will also need to consider the varietiesJ D ′ q1q2,D ′ q1q2 = J D ′ q1q2 (Γ 0 (N/D ′ ) ∩ Γ) andJ Dq1q2,Dq1q2 = J Dq1q2 (Γ 0 (N/D) ∩ Γ). LetÃ D ′ ,Ã D ,Ã D ′ q1q2 , andÃ Dq1q2 denote the categories of abelian varieties over Q which are Hecke-equivariantly isogenous toJ D ′ ,D ′ ,J D,D ,J D ′ q1q2,D ′ q1q2 , andJ Dq1q2,Dq1q2 , respectively. Also consider the corresponding categories (Ã D ′ ) old and (Ã D ) old associated to the q 1 q 2 -old quotients ofJ D ′ and J D , respectively. We adopt all of the notational conventions introduced in section 5 for these varieties as well; for instance,J D,Dq1q2 will denote the q 1 q 2 -new subvariety ofJ D,D . Finally, letm be a maximal ideal ofT D ′ whose associated Galois representation is isomorphic to ρ m .
The four degeneracy maps φ 1 , φ q1 , φ q2 , and φ q1q2 from where the subscript old denotes the q 1 q 2 -old subvariety. The action of T q1 and T q2 on (J D ′ ,D ′ ) 4 is given by the matrices: where τ q1 and τ q2 denote the endomorphisms of J D ′ ,D ′ induced by the q 1 th and q 2 th Hecke operators. The diamond bracket operators, and the Hecke operators T n for n prime to q 1 q 2 , act diagonally. By the same arguments as those that follow the proof of Lemma 7.1, we have In particular (T D ′ ) old is a free T D ′ -algebra of rank 4. More generally, given any abelian variety J in A D ′ or A D , we can form the abelian variety J 4 on which (T D ′ ) old acts via the matrices given above. In this situation, we have: Lemma 7.3 Let A 1 and A 2 be abelian varieties in A D ′ or A D . Then 1 → A 4 2 is given by a four by four matrix of maps A 1 → A 2 . For such a map to be (T D ′ ) oldequivariant, it must commute with the matrices defining the action of T q1 and T q2 . It is then straightforward to check that every such matrix can be written in the form a 4 + T q1 b 4 + T q2 c 4 + T q1 T q2 d 4 for maps a, b, c, and d from A 1 to A 2 . (Here a 4 : (A 1 ) 4 → (A 2 ) 4 is the map obtained by evaluating the map a at each entry of the four-tuple defining a point of (A 1 ) 4 .) That all maps of the above form commute with T q1 and T q2 is clear.
✷ Recall that we have fixed abelian varieties J min D ′ and J min D in A D ′ and A D which satisfy "multiplicity one" at all maximal ideals outside S D ′ or S D . We will use the above results to make "compatible" choices of varieties J min for each of the other categories involved. The key is the following lemma: Proof. The set {1, T q1 , T q2 , T q1 T q2 } is a basis for (T D ′ ) old over T D ′ . This basis gives rise to a dual basis for Hom T D ′ ((T D ′ ) old , T D ′ ), and the matrices giving the action of T q1 and T q2 with respect to this dual basis are the same as the matrices giving the action of T q1 and T q2 on ( ). The result follows immediately, and the proof for J D is exactly the same. ✷ It follows from this that (J min D ′ ) 4 satisfies "multiplicity one" at every non-Eisenstein maximal ideal of (T D ′ ) old of residue characteristic greater than 3. By Theorem 4.11, it is straightforward to check that there is an abelian varietyJ min D ′ ofÃ D ′ such thatJ min D ′ satisfies "multiplicity one" at every non-Eisenstein maximal ideal ofT D ′ of residue characteristic greater than 3, and such that (J min D ′ ) old = (J min D ′ ) 4 . As usual, we takẽ J min D = (J min D ′ ) pp ′ -new , and similarly forJ min D ′ q1q2 andJ min Dq1q2 . Finally, to (Ã D ′ ) old and (Ã D ) old we associatẽ J min D ′ ,old = (J min D ′ ) 4 andJ min D,old = (J min D ) 4 , respectively. For each categoryÃ in the set the above choice of abelian variety gives a functor [−]Ã fromÃ to the category of "rank one" modules over the corresponding Hecke algebra.
Lemma 7.5 In this setting we have:

Let
Proof. We have a natural map [J]Ã → [J old ]Ã old defined by restricting an element of [J] to its q 1 q 2 -old subvariety. Locally at anym outside SÃ, this map fits into a commutative diagram: Since J old is a subvariety of J, the right-hand vertical map is surjective. But TmJ miñ A old is free of rank two, so the natural map ([J]Ã)m → ([J old ]Ãold )m is surjective as well. Statement (1) thus follows by Lemma 5.1. Statement (2) is immediate from the previous lemma.
✷ With these technicalities out of the way, we begin our study of the geometry of these varieties. First of all, we have: Lemma 7.6 There are non-canonical isomorphisms: Proof. We construct these isomorphisms forJ D,Dq1q2 andJ Dq1q2,Dq1q2 ; the other case is nearly identical. SinceJ Dq1q2,Dq1q2 is self-dual, we have Sincem is controllable at q 1 , we have an isomorphism by Proposition 4.14 and Lemma 6.5. On the other hand, by Proposition 5.7, we have an isomorphism Combining this with the previous two isomorphisms we find an isomorphism Asm is controllable at q 2 , X q2 (J D,Dq1q2 )m is isomorphic to [J D,Dq1q2 ]m by Proposition 4.14 and Lemma 6.5. so it suffices to show that the opposite inequality holds. Let Y q1 denote the character group ofJ Dq1q2,Dq1q2 [m] at q 1 , and X q2 denote the character group of J D (Γ 0 (q 2 N/D) ∩ Γ) 2 at q 2 . Sincem is not Eisenstein and has residue characteristic greater than 3, Proposition 5.9 implies that (Y * q1 /Y q1 )m is isomorphic to (X q2 /(T 2 q1 − q 1 )X q2 )m. In particular, we have: We will relate Y * q1 /mY * q1 toJ Dq1q2,Dq1q2 [m], and X q2 /mX q2 toJ D,D [m]. The key is controllability ofm at q 1 and q 2 .
Combining this result with the above inequality, we see that it suffices to show that Sincem is controllable at q 2 , (X q2 ) 2 m is isomorphic to TmJ D (Γ 0 (q 2 N/D) ∩ Γ), by Corollary 4.15. Thus By [DT], Theorem 2, the isogeny Lemma 7.8 There is a map g in Hom(J Dq1q2,Dq1q2 ,J D,D )m whose dual g † fits in a commutative diagram: in which the horizontal maps are those in the exact sequence of Theorem 5.4, the left-hand vertical map is induced by g † , and the right-hand vertical map is induced by f † .
Proof ] whose cokernel is supported away fromm. By Theorem 4.11, this implies that there is an isogeny φ :J Dq1q2,Dq1q2 →J D,Dq1q2 whose kernel is supported away fromm. Then φ † is an isogeny from (J Dq1q2,Dq1q2 ) † to (J D,Dq1q2 ) † whose kernel is supported away fromm. ButJ Dq1q2,Dq1q2 is self-dual, and so we can consider φ † as an isogeny from (J D,D ) q 1 q 2 -new toJ Dq1q2,Dq1q2 , whose kernel is supported away from m. Viewed in this way, φ † induces isomorphisms between [J Dq1q2,Dq1q2 ]m and [(J D,D ) q 1 q 2 -new ]m, as well as between X p ((J D,D ) q 1 q 2 -new )m and X p (J Dq1q2,Dq1q2 )m. Under these identifications, the natural map: is identified with the map: It thus suffices to show that the latter is an isomorphism. We have a sequence of isomorphisms: (The first isomorphism comes from Proposition 4.14; the second from the adjointness of Hom and tensor product, and the third from Corollary 6.6, using the fact thatm is controllable at q 1 .) It follows by Lemma 6.2 that the natural map in question is an isomorphism. (and similarly forJ Dq,Dq ,) it follows that the right-hand vertical map becomes an isomorphism after tensoring withT/m. Thus, after tensoring withT/m, the bottom horizontal map and the right-hand vertical maps are isomorphisms, and the left-hand vertical map is surjective. It follows that the upper horizontal map becomes surjective after tensoring withT/m. By Nakayama's lemma, this implies that the upper horizontal map is surjective; since both of the modules in the top row have the same Z l -rank and no Z l -torsion, the upper horizontal map is an isomorphism. The result follows immediately. We have a commutative diagram: in which the vertical maps are induced by the surjectionJ D,D → (J D,D ) old , and the horizontal maps are the Z l -duals of the maps taking a morphism of abelian varieties to the induced maps on character groups. By the previous proposition, the upper horizontal map is an isomorphism; the right-hand vertical map is surjective. It follows that the lower horizontal map is surjective, and hence an isomorphism. ✷ The following technical lemma is now all that we need in order to establish Theorem 6.1 at m.
on the latter, and via the transposes of these matrices on the former. After tensoring with Z l , conjugation with the diagonal matrix sends the above matrices to their transposes. Thus P induces an isomorphism of (T D ) * ⊗T D ⊗ Z l with (T D ) * ⊗ Z l . Localizing atm then gives the desired isomorphism. Now we have: The is an isomorphism. By Lemma 7.11 and Proposition 7.10, the natural map: is an isomorphism. It is easy to check that this latter map is just the corresponding map

Main results
Having successfully proven Theorem 6.1, we will now make use of it to compute [J D ] for D dividing N .
Recall that for each D dividing N with an even number of prime divisors, we have constructed a locally-free Proposition 8.1 Let p and q be two primes dividing D. Then we have Proof. We have a sequence of isomorphisms: which immediately proves the claim. ✷ To extend this to a relationship between [J D,R ] and [J Dpq,R ] for R divisible by Dpq, we use the following lemma: Lemma 8.2 Let D R, and R ′ be divisors of N such that D is divisible by an even number of primes, and such that D|R|R ′ . Then [J D, Proof. There is a natural map [J D,R ] → [J D,R ′ ] given by taking an element of [J D,R ], treating it as an element of Hom(J D,R , J min R ), and restricting it to J D,R ′ ⊂ J D,R . (The image of J D,R ′ will always land in J min R ′ ⊂ J min R .) By Lemma 5.1, it suffices to show the surjectivity of this map at m outside S R ′ . Letm be the preimage of m in T R . We have a commutative diagram: in which the vertical maps are isomorphisms. The lower horizontal map is induced by an inclusion of varieties and is therefore surjective. The upper horizontal map is the restriction map we wish to study, tensored with TmJ min R . The surjectivity of the lower horizontal map means that the upper horizontal map is surjective as well. But since TmJ min R is free of rank two over T R m , this implies that the map [J D,R ]m → [J D,R ′ ] m is surjective as well.
✷ We also need to relate L R with L R ′ for R dividing R ′ ; since we have chosen our J min R compatibly, we have the following: Dualizing, and invoking Corollary 5.2, we find that L −1 R ′ ∼ = L R ⊗ T R T R ′ , as required.
✷ Putting these together, and passing to A N , we find: Proposition 8.4 Let D divide N and be divisible by an even number of primes. For any p and q dividing D, we have: Proof. Tensor the isomorphism of Proposition 8.1 with T N and apply Lemmas 8.2 and 8.3. ✷ Working inductively, one immediately proves: Theorem 8.5 For any D dividing N and divisible by 2k primes, we have: Combining this with Proposition 4.12, we immediately obtain: Corollary 8.6 Let D and D ′ be divisors of N , divisible by 2n and 2m primes, respectively, such that D divides D ′ . Let M be the module Then there is an S-isomorphism: This result determines the S-isomorphism class of J D ′ in terms of J D ; in particular, if one has J D , and one wishes to construct an abelian variety S-isomorphic to J D ′ , one merely has to find an ideal I of T (or, more generally, of End(J D )) isomorphic to M , and construct J D /J D [I]. Unfortunately, although M is independent of the choice of J min , the various factors in the product defining M are not, making M difficult to compute. One can remedy this, however, by introducing a suitable "multiplicity one" hypothesis.
In particular, we expand our set S to include those maximal ideals for which J 1,1 = J 1 (Γ 0 (N ) ∩ Γ) fails to satisfy "multiplicity one"; that is, we take S R + to be the set of m ⊂ T R such that either m is in S R or J 1 (Γ 0 (N ) ∩ Γ)[m] has dimension greater than two. (The latter is expected to be a very rare occurrence; in particular it cannot occur if l is prime to 2N M [Ti]. Moreover, the only known counterexamples are in residue characteristic 2 [Ki].) As usual we take S + = S N + . The advantage of enlarging our set of "bad primes" to S + is that we can take J min R = J 1,R for all R; the "multiplicity one" condition we have imposed above makes this a valid choice. Moreover, we can compute L N in this setting. Since J 1,1 We thus recover the result that T 1 S 1 + is Gorenstein, and find that L 1 ∼ = S 1 + ((T 1 S 1 + ) * ) −1 . Lemma 8.3 then determines L N explicitly. Putting this together, we find: Theorem 8.7 For any D dividing N and divisible by exactly 2k primes, we have: Corollary 8.8 Let D and D ′ be divisors of N , divisible by exactly 2n and 2m primes, respectively, such that D divides D ′ . Let M be the module Then there is an S + -isomorphism: Remark 8.9 The previous two results, although valid over a smaller subset of Spec T, have the advantage that the character groups that appear within them are explicitly computable, via the techniques in [Ko] together with Corollary 5.3. It is thus possible, given the isomorphism class of J 1 , to effectively compute an abelian variety which is S + -isomorphic to J D for any D one desires.
The following result, a strengthening of a result proved by Yang [Ya] in the case in which ρ m is ramified at at least half of the primes dividing D, is an immediate corollary.
If m is controllable at p, X p (J min ) m is free of rank one, and can be ignored. If m is not controllable at p, the surjection T m J min → X p (J min ) m shows that X p (J min )/mX p (J min ) has dimension at most two. In particular, X p (J min ) m is generated by a two-element set, so tensoring with X p (J min ) m increases the size of a minimal generating set by at most a factor of two. The result follows. ✷ Considering the special case in which m is controllable at every prime dividing D yields a "multiplicity one" result for Jacobians of Shimura curves: . If m satisfies "multiplicity one" for the full Jacobian J 1,1 , then it also satisfies "multiplicity one" for J 1 , as the latter is just the N -new subvariety of the former. In particular J 1 [m] has dimension two and the result follows. ✷ Remark 8.12 Although we have been working throughout with maximal ideals m outside S (and hence of residue characteristic 2 or 3), the preceding corollary holds in more generality. In particular, if Γ = Γ 0 (M ) or if Γ contains Γ 1 (r), for r ≥ 4, then the preceding corollary holds for all non-Eisenstein m. To establish this, one observes observe that the only places in the above argument in which we need the maximal ideals of residue characteristic 2 or 3 to lie in S are in Proposition 5.6 and in the level-raising arguments of Section 7. Since the latter are unnecessary for an m which is already controllable at every prime dividing D, and Proposition 5.6 holds even in residue characteristics 2 and 3 when Γ = Γ 0 (M ) or Γ 1 (M ) [Ri4], all of the results we have obtained above hold locally at m. This is enough to establish the "multiplicity one" result above.

Further questions
These results leave several questions unanswered. The most obvious of these is whether or not the set S can be made smaller. At the moment, a maximal ideal m can be in S either because it is Eisenstein or because it has residue characteristic 2 or 3. It seems likely that in many cases the non-Eisenstein maximal ideals of residue characteristic 2 and 3 may be removed from S. At the moment there are two places in the argument that require the residue characteristic to be greater than 3. The first is that component groups of Jacobians of some Shimura curves can have support at non-Eisenstein primes of residue characteristic 2 or 3, which interferes with the proof of Proposition 5.7. This does not happen, however, if Γ either has the form Γ 0 (M ) for some M , or if Γ contains Γ 1 (r) for some r ≥ 4 [Ri4]. Thus in many cases this does not pose a problem.
The more serious difficulty occurs in Section 7, where we rely on results of [DT] which only hold in residue characteristic greater than 3. The only result which we really need from that paper is Theorem 2 (we invoke Theorem A at one point but if one has Theorem 2 then the case we need of Theorem A seems to follow quickly.) Moreover, since we have the luxury of choosing the primes at which we raise the level, we need only find two primes q 1 and q 2 for which Theorem 2 holds locally at m. It seems very unlikely that this will not happen. It is thus reasonable to conjecture that (when Γ satisfies one of the conditions given above) Theorem 6.1 holds even when S contains only Eisenstein primes.
The non-Eisenstein condition, on the other hand, is much more serious. All of the techniques of section 4, and Proposition 5.7, fail to hold at Eisenstein maximal ideals. Thus applying the techniques of this paper in this case seems hopeless.
We have also left open the question of whether or not there exist canonical maps between the varieties in question. If one is interested in looking for such things, one place to start would be to try to construct canonical maps between the modules constructed in Section 8.
Finally, Corollary 8.11 provides a sufficient condition for "multiplicity one" for Jacobians of Shimura curves. It is interesting to ask if this condition is necessary as well. This would hold if one could establish a converse to Lemma 6.5. A special case of such a result appears in [Ri2], where Ribet constructs character groups which are not free of rank one at specific maximal ideals of the Hecke algebra.
4. At a supersingular point x of (X p ) Fp , the completion of the strict henselation of the local ring is isomorphic to W (F p )[ [X, Y ]]/(XY − p kx ), where k x is the order of the automorphism group of the false elliptic curve with level structure corresponding the the point x modulo {±1} if −1 is in Γ, and the order of the full automorphism group if −1 is not in Γ.
Proof. Statement 2 is well-known, and statement 1 is known except for the regularity of X p at nonsupersingular points on the characteristic p fiber. It thus suffices to prove this regularity, along with statements 3 and 4. The arguments we give are basically adaptations of arguments found in [DR] for the case when D = 1. Let r > 3 be a prime which does not divide N Dp, and consider the curves X r = X D (Γ ∩ Γ 1 (r)) and X p,r = X D (Γ 0 (p) ∩ Γ ∩ Γ 1 (r)). Then X r and X p,r are fine moduli spaces, so the above theorem applies. We have maps X r → X and X p,r → X p obtained by forgetting the level structure at r. These maps are finite, and ramified precisely at those x in X or X p for which k x > 1. Moreover, at any x in X (resp. X p ) for which k x > 1, the automorphism group of x acts on the local ring of any preimage of x in X r (resp. X p,r ). This action is faithful if −1 is not in Γ, and has kernel {±1} if −1 is in Γ. Let G x denote the group Aut(x) in the former case, and Aut(x)/{±1} in the latter. Then the above maps induce isomorphisms: wherex is any lift of x to X r or X p,r , respectively, andÔ X,x is the completion of the strict henselization of the local ring of X at x. Suppose x is a point on the characteristic p fiber of X p . Ifx is a regular point of X p,r , thenÔ Xp,r,x is isomorphic to W (F p )[ [X]]. By the same argument as the proof of [DR], VI.6.9, the subring of G x -invariants of this ring is the ring W (F p )[ [X kx ]]. In particular, x is a regular point. This proves statement 1.
Ifx is a supersingular point of X p,r , thenÔ Xp,r ,x is isomorphic to W (F p )[ [X, Y ]]/(XY − p). Again, the same argument as the proof of [DR], VI.6.9 shows that the subring of G x -invariants is generated by X kx and Y kx , and hence isomorphic to W (F p )[ [X, Y ]]/(XY − p kx ). This proves statement 4.
For statement 3, we have a commutative diagram (X r in which the horizontal maps can be either both Frob or both Ver. Since the map from X p,r to X p is a surjection, X p will thus be the union of the images of Frob and Ver, and these images will intersect precisely at the supersingular points of X p . All that remains is to prove that Frob and Ver are closed immersions. They are clearly injections on closed points. Thus, since X is proper, to prove they are closed immersions it suffices to show that they are injections on tangent spaces. First observe that for any x ∈ X, G x = G Frob(x) , since any automorphism of a false elliptic curve commutes with Frobenius. Thus if we fix a liftx of x to X r , then G x acts on botĥ O Xr ,x andÔ Xp,r ,Frob(x) . Moreover, the map fromÔ Xp,Frob(x) toÔ X,x induced by Frob is just the restriction of the corresponding map fromÔ Xp,r ,Frob(x) toÔ Xr ,x to the ring of G x -invariants ofÔ Xp,r ,Frob(x) , by the commutative diagram given above. In particular, our explicit computation of these rings shows that it is surjective. Thus Frob induces an injection on tangent spaces, and is therefore a closed immersion. The same argument shows that Ver is a closed immersion. ✷