ISRAEL JOURNAL OF MATHEMATICS 189 (2012), 1-38 DOI: 10.1007/s11856-011-0172-8

RIBET BIMODULES AND THE SPECIALIZATION OF HEEGNER POINTS

Santiago Molina*

Matemática aplicada IV, Universitat Politécnica de Catalunya Vilanova i la Geltru, Barcelona, Spain e-mail: santimolin@gmail.com

ABSTRACT

For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = Xo(D,N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P £ CM(R) specializes to a singular point (resp. a irreducible component) of the special fiber X of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of Xo(D, N) and K. Ribet's theory of bimodules, we give a construction of a correspondence $ between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebra that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin—Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in X. We also illustrate our results with an explicit example.

1. Introduction

Let B be a quaternion algebra over the field Q of rational numbers, and let D > 1 denote its reduced discriminant. Let N > 1 be a positive integer, coprime to D, let £(D, N) stand for the set of oriented Eichler orders of level

* The research of the author is supported financially by DGICYT Grant MTM2009-13060-C02-01.

Received February 3, 2010 and in revised form April 2, 2011

N in B (cf. §2.1 for precise definitions) and finally let Pic(D, N) denote the set of isomorphism classes of such orders.

For any given O G E(D,N), it is well-known that Pic(D,N) is in bijective correspondence with the set of classes of projective left O-ideals in B up to principal ideals.

If B is indefinite, i.e., D is the (square-free) product of an even number of primes, then Pic(D, N) is trivial for any N; if B is definite, Pic(D, N) is a finite set whose cardinality h(D, N) is often referred to as the class number of O.

Let K/Q be an imaginary quadratic number field and let R be an order of K of conductor c. For any O G E(D, N) let CMo(R) denote the set of optimal embeddings ^ : R — O, up to conjugation by Ox. Define

CMd,n (R) = L_|CM0 (R),

where O G E(D, N) runs over a set of representatives of Pic(D, N). Write

n : CMDN(R) —> Pic(D, N)

for the natural forgetful projection which maps a conjugacy class of optimal embeddings ^ : R--O to the isomorphism class of its target O.

Let Xo(D, N )/Q denote Shimura's canonical model of the coarse moduli space of abelian surfaces with multiplication by an Eichler order O of level N in B. As is well-known, associated to each R as above there is a (possibly empty) set of algebraic points CM(R) C X0(D,N)(Kab), which is in natural one-to-one correspondence with CMo (R). These points are called special, CM or Heegner points in the literature. Below we shall recall natural actions of the class group Pic(R) and of the Atkin-Lehner group W(D, N) on both sets CMo (R) and CM(R) which are intertwined by this correspondence. In the case of Pic(R) this is the statement of Shimura's reciprocity law. For technical reasons we are going to assume for the rest of the paper that (c, DN) = 1.

Let now p be a prime, p \ c. It follows from the work of various people, including Deligne-Rapoport, Buzzard, Morita, Cerednik and Drinfeld, that X0(D, N) has good reduction at p if and only if p \ DN. There exists a proper integral model Xo(D, N) of Xq(D, N) over Spec(Z), smooth over which suitably

extends the moduli interpretation to arbitrary base schemes (cf. [2], [19]). When p is fixed in the context and no confusion can arise, we shall write X0(D,N) for its special fiber at p. It is known that each of the sets S of

• Singular points of X0(D, N) for p || DN,

• Irreducible components of X0(D, N) for p || DN,

• Supersingular points of X0(D, N) for p \ DN

are in one-to-one correspondence with either one or two copies of Pic(d, n) for certain d and n (cf. §5, 6, 7 for precise statements).

In each of these three cases, we show that under appropriate behavior of p in R C K (that we make explicit in §5, 6, 7, respectively—see also Table 1), Heegner points P G CM(R) specialize to elements of S (with the obvious meaning in each case).

This yields a map which, composed with the previously mentioned identifications CMAW(R) = CM(R) C X0(D, N)(Kab) and S = |j=1 Pic(d, n) (t = 1 or 2), takes the form

(1.1) CMd,n(R) |_| Pic(d, n).

Our first observation is that, although the construction of this map is of geometric nature, both the source and the target are pure algebraic objects. Hence, one could ask whether there is a pure algebraic description of the arrow itself.

One of the aims of this note is to exploit Ribet's theory of bimodules in order to show that this is indeed the case, and that in fact (1.1) can be refined as follows: there exists a map $ : CM^,»(R) —> Ut=1 CMd,n(R) which is equivariant for the actions of Pic(R) and of the Atkin-Lehner groups and makes the diagram

(1.2) CMDtN(R) ULi CMdi„(R)

U,4=1 Pic (d,n)

commutative (see (5.10), (5.12), (6.13), (7.14) and (7.15)). Moreover, we show that when N is square free, the maps $ are bijections.

As we explain in this article, the most natural construction of the map $ is again geometrical, but can also be recovered in pure algebraic terms.

The main bulk of the article is devoted to the case p | D; the other cases are comparatively much simpler and probably already known, at least partially.

Table 1

p e cm (R) Condition on p S'=Irreducible components <¡>

p regular V 1 D p inert in K U2=1Pic (f,JV) CMD,jv(ß) ^ u|=1 CM£ N(R)

P II N p splits in K U?=1Pic(D,f) cmd¡n(R) ^ U?=1CMD n(R)

P singular Condition on p S'=Singular Points

V 1 D p ramifies in K Pic(f ,Np) CMD,(v(ß)^CMi, (R)

P II N p ramifies in K Pi <Dp,f) cmd,jv(ß)^cm jv(ß) F, p

For the convenience of the reader, we summarize the situation for p || DN in Table 1. Here P stands for the specialization of the point P G CM(R).

Let us summarize here a weak, simplified version of some of the main results of this note, Theorems 5.3, 5.4, 5.8 and 7.3. We keep the notation as above.

Theorem 1.1: (1) Let p | D.

(i) A Heegner point P G CM(R) of X0(D,N) reduces to a singular point of X0(D, N) if and only if p ramifies in K.

(ii) Assume p | disc(K) and N is square-free. Then there is a one-to-one correspondence $ : CM(R) —^ CMD/pNp(R) which is equivariant for the action of Pic(R) and W(D,N).

(iii) Assume p \ disc(K) and N is square-free. Then there is a one-to-one correspondence $ : CM(R) ^ CMD/pN(R) U CMD/pN(R) which is equivariant for the action of Pic(R) and W(D, N).

(2) Let p | (N, disc(K)) and N be square-free. Then there is a one-to-one correspondence $ : CM(R) ^^ CMDp N/p(R) which is equivariant for the action of Pic(R) and W(D, N).

The above statements are indeed a weak version of our results, specially because we limited ourselves to claim the mere existence of the correspondences $, whereas in §5 and §7 we actually provide an explicit computable description of them in terms of bimodules. Combined with generalizations of the results of P. Michel [16, Theorem 10] admitting ramification at primes dividing D ■ N, we also obtain the following equidistribution result, which the reader may like to view as complementary to those of Cornut-Vatsal and Jetchev-Kane (cf. [16, page 190] for an analogous discussion on the supersingular situation).

Below, for any singular point s or irreducible component c of Xo(D, N), the weight (or length or thickness, in the literature) w of s or c is defined to be w := Ox |/2, where O is the oriented order attached to s or c by the above correspondences.

Corollary 1.2: Suppose that D N is square-free. Let p be a prime and write X0(D, N) for the special fiber at p of the Shimura curve X0(D, N).

(i) Assumep | DN andp | disc(K). Let X0(D, N)sing = {si, s2,..., sh} be theset of singular points ofX0(D,N) and let n : CM(R)^X0(D, N)sing denote the specialization map. Then, as disc(Kthe Galois orbits Pic(R) * Q, where R is the maximal order of K and Q G CM(R), are equidistributed in X0(D, N)sing relatively to the measure given by

More precisely, there exists an absolute constant n > 0 such that #{P G Pic(R) * Q : n(P) = si}

#CM(R)

M(sj)+ O(diso(^ )-n ).

(ii) Assumep D and is inert in K. Let X0(D,N)c ={c1,c2,...,ct} be the set of irreducible components of X0(D,N) and let nc : CM(R)^X0(D,N )c denote the map which assigns to a point P G CM(R) the irreducible component where its specialization lies. Then, as disc(Kthe Galois orbits Pic(R) *Q C CM(R), where R is the maximal order of K and Q G CM(R), are equidistributed in X0(D, N)c relatively to the measure given by

^(ci)=w(ci)-x j ^¿i w(cj ^.

More precisely, there exists an absolute constant n > 0 such that

= +o(disc(AT")

The original motivation of this research was the case of specialization of Heegner points P G CM(R) on Cerednik-Drinfeld's special fiber X0(D, N) at a prime p | D, particularly when p ramifies in R. There are several reasons which make this scenario specially interesting (beyond the fact that so far it had not

been studied at all, as most articles on the subject exclude systematically this case):

(1) Let f be a newform of weight 2 and level L > 1 such that there exists a prime p || L which ramifies in K. Assume L admits a factorization L = DN into coprime integers D and N where p | D, and D/p is the square-free product of an odd number of primes, none of which splits in K; all prime factors of N split in K. This is a situation in which little is known about the Birch and Swinnerton-Dyer conjecture for the abelian variety Af attached to f by Shimura, when looked over K and over its class fields.

It follows that for every order R in K of conductor c > 1 such that (c,L) = 1, CM(R) C X0(D,N)(Kab) is a nonempty subset of Heegner points. Points P G CM(R) specialize to singular points on the special fiber X0(D,N) at p; degree-zero linear combinations of them yield points on the Neron model J of the Jacobian of X0(D, N) over Zp, which can be projected to the group $p of connected components of J It is expected that the results of this paper will be helpful in the study of the position of the image of such Heegner divisors in $p in analogy with the work of Edixhoven in [1, Appendix] (though the setting here is quite different from his).

Moreover, if one further assumes that the sign of the functional equation of f is +1 (and this only depends on its behavior at the prime factors of D), it is expected that this can be related to the special values L(f/K, x, 1) of the L-function of f over K twisted by finite characters unramified at p, as an avatar of the Gross-Zagier formula in the spirit of [1]. If true, a system of Kolyvagin classes could be constructed out of the above-mentioned Heegner divisors and a word could be said on the arithmetic of Af. The author hopes to pursue these results in the work in progress [18].

(2) On the computational side, there is an old, basic question which seems to remain quite unapproachable: can one write down explicit equations of curves X0(D,N) over Q when D > 1? As is well-known, classical elliptic modular curves X0(N) := X0(1, N) can be tackled thanks to the presence of cusps, a feature which is only available in the case D = 1. Ihara [10] was probably one of the first to express an interest in this

problem, and already found an equation for the genus 0 curve Xo(6,1), while challenged to find others. Since then, several authors have contributed to this question (Kurihara, Jordan, Elkies, Clark-Voight for genus 0 or/and 1, Gonzalez-Rotger for genus 1 and 2). The methods of the latter are heavily based on Cerednik-Drinfeld's theory for X0(D, N) x Zp for p | D and the arithmetic properties of fixed points by Atkin-Lehner involutions on X0(D, N). It turns out that these fixed points are usually Heegner points associated to fields K in which at least one (sometimes all!) prime p | D ramifies: one (among others) of the reasons why the methods of [8], [7] do not easily extend to curves of higher genus is the little understanding one has of the specialization of these points on the special fiber X0(D,N) at p. It is hoped that this article can partially cover this gap and be used to find explicit models of many other Shimura curves: details for hyperelliptic curves may appear in [17]. For this application, our description of the geometrically-constructed maps $ in pure algebraic terms by means of Ribet's bimodules turns out to be crucial, as this allows one to translate it into an algorithm.

In the last section of this note, we provide a numerical example which illustrates some of the explicit computations one can perform by using the material developed in this work.

Notation. Throughout, for any module M over Z and for any prime p we shall write Mp = M Zp. Similarly, for any homomorphism x : M^N of modules over Z, we shall write xp : Mp^Np for the natural homomorphism obtained by extension of scalars. We shall also write Z to denote the profinite completion of Z, and M = M Z.

For any Z-algebra D, write D0 = D Q and say that an embedding ^ : Di ^ D2 of Z-algebras is optimal if ) HD2 = y(D\) in D°.

Acknowledgements. The author would like to thank Massimo Bertolini, Matteo Longo, John Voight and specially Josep González and Victor Rotger for their comments and discussions throughout the development of this paper. The author also thanks Xevi Guitart and Francesc Fite for carefully reading a draft of the paper and providing many interesting remarks. The author also appreciates the anonymous referee for some helpful comments. Finally, the author

gratefully acknowledges the generous help received from the Centre de Recerca Matemática of Bellaterra (Barcelona) during these last months.

2. Preliminaries

2.1. Quaternion orders and optimal embeddings. Let B be a quaternion algebra over Q, of reduced discriminant D. An order O in B is Eichler if it is the intersection of two maximal orders. Its index N in any of the two maximal orders is called its level.

An orientation on O is a collection of choices, one for each prime p | DN: namely, a choice of a homomorphism op : O — Fp2 for each prime p | D, and a choice of a local maximal order Op containing Op for each p | N. One says that two oriented Eichler orders O, O' are isomorphic whenever there exists an automorphism x of B with x(O) = O' such that op = o'p o x for p | D and Xp(Op) = O'p for p | N. Fix an oriented Eichler order O in E(D, N).

Definition 2.1: Let O G E(D,N) be an oriented Eichler order. For a projective left O-ideal I in B, let I *O be the right order O' of I equipped with the following local orientations at p | DN. Since I is locally principal, we may write Ip = Opap for some ap G Bp, so that O'p = a-1 Opap. For p | D, define

o'p(a-1xap) = op(x). For p | N, define O'p = a- 1Oap.

As in the introduction, let R be an order in an imaginary quadratic field K. Two local embeddings $p,yp : Rp — Op are said to be equivalent, denoted yp ~p $p, if there exists A G Op such that = AypA-1. Let us denote by mp the number of equivalence classes of such local embeddings. Note that, in case p | D or p || N, we have mp G {0,1, 2} by [27, Theoreme 3.1, Theoreme 3.2]. We say that two global embeddings : R-—O are locally equivalent if yp ~p for all primes p.

By [27, Theoreme 5.11], there is a faithful action of Pic(R) on CMD , N(R), denoted [J] * y for [J] G Pic(R) and y G CMD , N (R). It can be explicitly defined as follows: if y : R—O, let [J] *O = Oy(J) * O and let [J] * y : R— [J] *O be the natural optimal embedding

R<^ {x e B : OV(J)x COtp(J)}.

The action of Pic(R) preserves local equivalence, and in fact (2.3) #CMDiN(R) = h(R) J] mP,

where h(R) = #Pic(R) is the class number of R.

For each pn || DN, there is also an Atkin-Lehner involution wpn acting on CMD N(R), which can be described as follows. Let Po denote the single two-sided ideal of O of norm pn. Notice that Po * O equals O as orders in B, but they are endowed with possibly different local orientations. Given p G CMo(R) C CMDjN(R), wpn maps p to the optimal embedding wpn (p) : R ^ Po * O, where wpn (p) is simply p as ring homomorphism.

In the particular case p || DN and mp = 2, the involution wp switches the two local equivalence classes at p.

For any m || DN we denote by wm the composition wm = Wpn\\m wpn. The set W(D, N) = {wm : m || DN} is an abelian 2-group with the operation wmwn = wnm/(m n)2, called the Atkin-Lehner group. Attached to the order R, we set

D(R) = IT p, N(R)= n P,

p\D,mP = 2 pn\N,mP = 2

Wd,n(R) = {wm G W(D, N) : m || D(R)N(R)}. Suppose that N is square-free. Since Pic(R) acts faithfully on CMD N(R) by preserving local equivalences, and Atkin-Lehner involutions wp G W(R) switch local equivalence classes at p, it follows from (2.3) that the group WD,N(R) x Pic(R) acts freely and transitively on the set CMDjN(R).

2.2. Shimura curves. Assume, only in this section, that B is indefinite. By an abelian surface with quaternionic multiplication (QM) by O over a field K, we mean a pair (A, i) where:

i) A/K is an abelian surface.

ii) i : O ^ End(A) is an optimal embedding.

Remark 2.2: The optimality condition i(B) n End(A) = i(O) is always satisfied when O is maximal.

For such a pair, we denote by End(A, i) the ring of endomorphisms which commute with i, i.e., End(A, i) = G End(A) : $ o i(a) = i(a) o $ for all a G O}.

Let us denote by X0(D,N)/Z Morita's integral model (cf. [2], [19]) of the Shimura curve associated to O. As Riemann surfaces, X0(D, N)C = r0(D, N)\H if D > 1; X0(D,N)C =r0(D,N)\(HUP1(Q)) if D = 1. Fixing an isomorphism B ® R ^ M2(R), the moduli interpretation of X0(D,N) yields a one-to-one correspondence

where i = iT arises from the natural embedding O ^ B ^ M2(R) and AT = i(O) ■ v for v = ( 1 ).

Above, two pairs (A, i) and (A', i') are isomorphic if there is an isomorphism $ : A —> A' such that $ o i(a) = i'(a) o $ for all a GO. Throughout, we shall denote by P = [A, i] the isomorphism class of (A, i), often regarded as a point on X0(D,N).

A (non-cusp) point P = [A, i] G X0(D, N)(C) is a Heegner (or CM) point by R if End(A, i) ~ R. We shall denote by CM(R) the set of such points. By the theory of complex multiplication, we actually have CM(R) C X0(D, N)(Q). As is well-known, there is a one-to-one correspondence

yP : R ~ End(A,i) — Endo(H1 (A, Z)) = O.

Throughout, we shall fix the isomorphism R ~ End(A, i) to be the canonical one described in [12, Definition 1.3.1].

Let P = [A, i] = [AT, iT] G CM(R). Since (c, DN) = 1, via yP we may regard O as a locally free right R-module of rank 2. As such,

O -R © eI,

for some e G B and some locally free R-ideal I in K. This decomposition allows us to decompose

ro (D,N )\H

Abelian surfaces (A, i)/C with quaternionic multiplication by O

[At = C2/At, iT]

CM(fi) 4 CMAW (R) P = [(A, ¿)] 4 fP

AT = i(O)v = i(fpR)v ® e ■ i(fpI)v.

Since R = End(A, i) and i(O) ( R = M2(R), it follows that

i(ppk) c c d4s m2(c).

Hence AT = i(pPR)v © i(pPI)ev and A is isomorphic to the product of two elliptic curves with CM by R, namely E = C/R and Ej = C/I. Moreover, the action of O on E x Ej induces the natural left action of O on R © el.

3. Bimodules: Ribet's work

3.1. Admissible bimodules. Let Z denote either Z or Zp for some prime p. Let O be an Eichler order and let S be a maximal order, both over Z, in two possibly distinct quaternion algebras B and H. By a (O, S)-bimodule M we mean a free module of finite rank over Z endowed with structures of left projective O-module and right (projective) S-module. For any (O, S)-bimodules M and N, let us denote by Homf, (M, N) the set of (O, S)-bimodule homomorphisms from M to N, i.e., Z-homomorphisms equivariant for the left action of O and the right action of S .If M = N, we shall write EndO (M) for HomO(M, M).

Notice that, since S is maximal, it is hereditary and thus all S-modules are projective. Note also that (O, S)-bimodules are naturally O (S-modules.

Let E be the (possibly empty) set of prime numbers which ramify in both O and S. For p in E, let Po and Ps denote the unique two-sided ideals of O and S, respectively, of norm p.

Definition 3.1: An (O, S)-bimodule M is said to be admissible if PoM = MPs for all p G E.

Remark 3.2: Let M be a (O, S)-bimodule of rank 4n over Z, for some n > 1. Since M is free over Z, M is also free over S as right module, by [6]. Up to choosing an isomorphism between M and Sn, there is a natural identification Ends (M) — Mn(S). Thus, giving a structure of left O-module on M amounts to giving a homomorphism f : O ^ Mn(S).

Since Eichler orders are Gorenstein (cf. [11]), the O-module M is projective if and only if O is precisely its left order; or equivalently f is optimal, f (B) n Mn(S) = f (O). In particular, we conclude that the isomorphism class of a (O, S)-bimodule M is completely determined by the GLn(S)-conjugacy class of an optimal embedding f : O ^ Mn (S). Finally, in terms of f, M is admissible if and only if f (PO) = Mn(PS) for all p G E.

We now proceed to describe Ribet's classification of -local and global- admissible bimodules. Let p be a prime. Let O denote the maximal order in a quaternion division algebra B over Qp. Let p = O • n be the maximal ideal of O and let Fp2 be the residue field of p.

Theorem 3.3 ([23, Theorem 1.2]): Assume Z = Zp, B is division over Qp and O is maximal in B. Let M be an admissible (O, O)-bimodule of finite rank over Zp. Then

M = Ox ■■■xOx p x ■■■ xp,

s-v-' >-v-'

r factors s factors

regarded as a bimodule via the natural action of O on O itself and on p given by left and right multiplication. In that case, we say that a (O, O)-bimodule M is of type (r, s)

Using the above local description, Ribet classifes global bimodules of rank 8 over Z in terms of their algebra of endomorphisms, provided O is maximal. Hence, assume for the rest of this subsection that Z = Z and O is maximal in B. Write DB = disc(B), DH = disc(H) for their reduced discriminants. Let M be a (O, S)-bimodule of rank 8, set

DM = n P

p\Ds Dh ,p

and let C be the quaternion algebra over Q of reduced discriminant Dq4 . Note that the class of C in the Brauer group Br(Q) of Q is the sum of the classes of the quaternion algebras B and H and, in fact, neither C nor DM4 depend on M, they only depend on B and H.

We shall further assume for convenience that Dq4 = 1, i.e., B ^ H.

Proposition 3.4 ([23, Proposition 2.1]): EndO(M) ® Q ~ C and the ring EndO (M) is an Eichler order in C of level]] k Pk, where pk are the primes in E such that Mpk is of type (1,1).

Assume now that O and S are equipped with orientations. According to [23, §2], the Eichler order A = Endf, (M) is endowed with orientations in a natural way, and the isomorphism class of M is determined by the isomorphism class of A as an oriented Eichler order.

Let Nq be the product of the primes pk in E such that Mpk is of type (1,1). Recall that rp + sp = 2, so the only possible types are (2,0), (1,1) and (0, 2).

Theorem 3.5 ([23, Theorem 2.4]): The map M — A induces a one-to-one correspondence between the set of isomorphism classes of admissible rank-8 bimodules of type (rp, sp) at p G and the set Pic(DQQ ,N0^) of isomorphism classes of oriented Eichler orders.

3.2. Supersingular surfaces and bimodules. Let p be a prime and let F be a fixed algebraic closure of Fp. An abelian surface A/F is supersingular if it is isogenous to a product of supersingular elliptic curves over F. Given a supersingular abelian surface A, one defines Oort's invariant a(A) as follows. If A is isomorphic to a product of supersingular elliptic curves, set a(A) = 2; otherwise set a(A) = 1 (see [14, Chapter 1] for an alternative definition of this invariant).

Let B be an indefinite quaternion algebra over Q, of reduced discriminant D. Let N > 1, (N,pD) = 1 and let O G E(D,N).

In this subsection we shall consider pairs (A, i) of supersingular abelian surfaces with QM by O over F such that a(A) = 2. By a theorem of Deligne (cf. [25, Theorem 3.5]) and Ogus [21, Theorem 6.2], A = E x E, where E is any fixed supersingular elliptic curve over F.

Let us denote by S = End(S) the endomorphism ring of E. According to a well-known theorem of Deuring [4], S is a maximal order in the quaternion algebra H = S® Q, which is definite of discriminant p. Therefore, giving such an abelian surface (A, i) with QM by O is equivalent to providing an optimal embedding

E: O M2(S) - End(A), or, thanks to Remark 3.2, a (O, S)-bimodule M of rank 8 over Z.

Remark 3.6: The maximal order S comes equipped with a natural orientation at p, and therefore can be regarded as an element of E (p, 1). This orientation arises from the action of S on a suitable quotient of the Dieudonne module of the elliptic curve E, which turns out to be described by a character S — Fp2. See [23, §4] for more details.

Let M = M(ji) be the bimodule attached to the pair (A, i) by the above construction. Then

EndO (M) -{7 G M (2, S) - End(A) | 7 o 1(a) = E(a) o 7, for all a GO} (3.5) =End(A,T).

The above discussion allows us to generalize Ribet's Theorem 3.5 to (O, S)-bimodules where O is an Eichler order in B of level N, (N,p) = 1, not necessarily maximal. Keep the notation E, DM and NM as in §3.1.

Theorem 3.7: The map M ^ Endf,(M) induces a one-to-one correspondence between the set of isomorphism classes of admissible (O, S)-bimodules M of rank 8 over Z and of type (rp, sp) at p G E, and the set Pic(DM, NNM) of isomorphism classes of oriented Eichler orders.

Proof. Given M as in the statement, let (A, i) be the abelian surface with QM by O over F attached to M by the preceding discussion. As explained in Appendix A, there is a one-to-one correspondence between isomorphism classes of such pairs (A, i) and triples (Ao, io, C), where (Ao, io) has QM by a maximal order O0 DO and C is a r0(N )-level structure.

Using Theorem 3.5, Ribet proves that End(A0, i0,C) C End(A0,i0) is an Eichler order of level NN0 and that such triples are characterized by their class in Pic(DQ, NN0q). We refer the reader to [23, Theorem 4.15] for p | D and to [23, Theorem 3.4] for p \ D.

Finally, End(A, i) = End(A0, i0, C) by Proposition 8.1 in Appendix A. This yields the desired result. I

4. Supersingular specialization of Heegner points

As in the previous section, let p be a prime and let F be a fixed algebraic closure of Fp. Let B be an indefinite quaternion algebra over Q of discriminant D. Let N > 1, (D,N) = 1 and O e S(D,N). Let P = [A,i} e X0(D,N)(Q) be a (non-cusp) point on the Shimura curve X0(D, N). Pick a field of definition M

of (A, i).

Fix a prime p of M above p and let A denote the special fiber of the Neron model of A at p. By [22, Theorem 3], A has potential good reduction at p. Hence, after extending the field M if necessary, we obtain that A is smooth over F.

Since A has good reduction at p, the natural morphism

0 : End(A) End(A)

is injective. The composition i = 0 o i yields^n optimal embedding i : O ^ End(A) and the isomorphism class of the pair (A, i) corresponds to the reduction

P of the point P on the special fiber X0(D, N) of X0(D, N)/Z at p.

Let us denote by $P : End(A, i) — End(A, i) the restriction of $ to End(A, i).

Lemma 4.1: The embeddings $ and $P are optimal locally at every prime but possibly at p.

Proof. We first show that $£ is optimal for any prime I = p. For this, let us identify End0(A)£ = End(A)£®Q with a subalgebra of End0(A)£ = End(A)£®Q via $, so that we must prove that End°(A)£ n End(A)^ = End(A)^. It is clear that End0(A)^ n End(A)^ D End(A)^. As for the reversed inclusion, let a End°(A)£ n End(A)^ and let M be a field of definition of a. Due to good reduction, there is an isomorphism of Tate modules T?(A) = T?(A). Falt-ings's theorem on Tate's Conjecture [26, Theorem 7.7] asserts that EndM(A)^ = End^M(Ti(A)) C End(T?(A)), where GM stands for the absolute Galois group of M and EndcM (T^(A)) is the subgroup of End(T^(A)) fixed by the action of GM. Since a G EndM (A)^, there exists n G Z such that na G EndM (A)^ = EndGM(Te(A)). By hypothesis a G EndF(A)£ C End(T^(A)), hence it easily follows that a G EndGM(Te(A)) = EndM(A)e C End(A)£.

This shows that $£ is optimal. One easily concludes the same for ($P)i by taking into account that End(A, i) = End(A, i) fl End(A). I

Remark 4.2: Notice that if End(A, i) is maximal in End0(A, i), the embedding $p is optimal. In fact, if End(A, i) is not maximal in End0 (A, i) the embedding $P may not be optimal at p. For example, let R be an order in an imaginary quadratic field K of conductor cpr, let A — E x E where E is an elliptic curve with CM by R and let i : M2 (Z) — M2(R) = End(A). Then End(A, i) — R whereas, if p is inert in K, End(A, i) G E(p, 1). Thus if $P was optimal, it would provide an element of CMp ,i(R) which is impossible by [27, Corollaire 5.12].

Let P = [A, i] G CM(R) C X0(D, N)(C) and denote by c the conductor of R. In §2.2 we proved that A = E x Ei where E = C/R, Ei = C/I are elliptic curves with CM by R. Here I is a projective R-ideal in K.

Assume that p is coprime to cN and does not split in K. Then A — E x Ei is a product of supersingular elliptic curves over F. Let S G E (p, 1) be the endomorphism ring of EE endowed with the natural orientation described in §3.2. Since a(A) = 2, we can assign a (O, S)-bimodule M = Mp to P = [A, i] as in the previous section.

Theorem 4.3: (a) There exists an optimal embedding ^ : R — S such

that, for all P G CM(R), (4.6) Mp — O®R S,

where S is regarded as left R-module via ^ and O as right R-module via y p.

(b) Upon the identifications (3.5) and (4.6), the optimal embedding $P is given by the rule

R — EndO (O®r S) 5 ——> $P (5): a ® s — aS ® s,

up to conjugation by Endf,(O ®R S)x.

Proof. As explained in §3.2, the isomorphism class of the bimodule Mp is completely determined by the optimal embedding i : O —> End(A) = M2(S), which in turn is defined as the composition of i with $ : End(A) — End(A).

More explicitly, as described in §2.2, the action of O on A — E x Ei given by the embedding i : O — End(A) — End(E x Ei) can be canonically identified with the action of O on the right R-module O — R © eI given by left multiplication (a,x) — a • x, where here O is regarded as a right R-module via yP. Therefore, the embedding i : O —> M2 (S) is the one induced by the action of O given by left multiplication on the S-module (R © eI) ®R S = O ®R S, where S is viewed as a left R-module via ^ : End(E) — R — End(i?) — S. This shows that Mp = O®R S.

Finally, since the action of 5 G R — End(E xEi,i) on E xEi is naturally identified with the action of 5 on O — R © eI given by right multiplication, the optimal embedding R — Endf, (O ®R S) arising from $P : End(A, i) — End(A, i) is given by $P(5)(a ® s) = aS ® s. This endomorphism clearly commutes with both (left and right) actions of O and S. Moreover, this construction is determined up to isomorphism of (O, S)-bimodules, i.e., up to conjugation by End$(Mp)x. ■

4.1. The action of Pic(R).

Definition 4.4: For a (O, S)-bimodule M, let

• Picg (M) denote the set of isomorphism classes of (O, S)-bimodules that are locally isomorphic to M,

• CMm (R) denote the set of End^ (M)x-conjugacy classes of optimal embeddings p : R — Endf, (M),

• CMMs(R) = {(N,V) : Ne PicO(M), ^ e CM^(R)}.

Let P = [A, i] e CM(R) be a Heegner point and let Mp be the (O, S)-bimodule attached to its specialization P = [A, i] at p as described above.

By Theorem 4.3, the bimodule Mp and the EndO(Mp)x-conjugacy class of $p: R— EndO (Mp) are determined by the optimal embedding pp e CMd,n(R). Hence specialization at p induces a map

(4.7) $ :CMd,n(R)^LI CMo!s(R), Pp—(Mp,$p),

where M runs over a set of representatives of local isomorphism classes of (O, S)-bimodules arising from some pP e CMD,N(R). Note that composing $ with the natural projection

(4.8) n ^CMMMs(R)^LI PicO(M), (M,ip)-M,

one obtains the map pP — Mp which assigns to pP the bimodule that describes the supersingular specialization of the Heegner point associated to it.

For any locally free rank-1 left O-module I, let us consider the right O-module

I-1 = {x e B : Ix C O}.

It follows directly from Definition 2.1 that I *O coincides with the left order of I- 1 , endowed with the natural local orientations.

Lemma 4.5: There is an isomorphism of (I *O, I *O)-bimodules between I *O and I-1 ®q I.

Proof. Since I *O = {x e B : Ix C I} = {x e B : xI-1 C I-1}, both I *O and I-1 ®q I are (I *O,I * O)-bimodules with the obvious left and right (I * O)-action. Moreover, by definition,

I-1 ®O I = {x e B ®O I = B : Ix CO®O I = I} = I *O,

which proves the desired result. I

Theorem 4.6 ([23, Theorem 2.3]): Let M be a (O, S)-bimodule and let A = End^ (M). Then the correspondence N — Hom^ (M, N) induces a bijection between Pic^ (M) and the set of isomorphism classes of locally free rank-1 right A-modules.

The one-to-one correspondence is given by

N I(N) = HomO (M, N))

N (I ) = I <a M <— I.

We can define an action of Pic(R) on CMMS(R) which generalizes the one on CMDN(R). Let (N,' : R ^ A) G CM^s(R) where N G PicO(M) and A = EndO (N ). Pick a representant J of a class [J] G Pic(R). Then '(J X)A is a locally free rank-1 right A-module. Write

[J] *N := '(J-1)A N G PicO(M)

for the element in Picf, (M ) corresponding to it by the correspondence of Theorem 4.6. Note that this construction does not depend on the representant J, since [J] *N only depends on the isomorphism class of the rank-1 right A-module '(J -1)A. Since R acts on '(J -1)A, there is an action of R on '(J -1)A <a N which commutes with the actions of both O and S. This yields a natural embedding [J] * ' : R^ EndO ([J] * N), which is optimal because

{x G K : '(x)('(J-1)A <8>a M) Ç '(J-1 )A ®a N} = R,

and does not depend on the representant J of [J]. Hence, it defines an action of [J] G Pic(R) on (N, ') G CMMS(R). Namely [J] * (N, ') = ([J] *N, [J] * ') G CMMs(R).

Given that both sets CM^,w (R) and CMMS(R) are equipped with an action of Pic(R), it seems reasonable to ask about the behavior of the action of Pic(R) under the map 0 : CMDN(R)^ UM CM^S(R) of (4.7). This is the aim of the rest of this subsection.

Recall that, B being indefinite, the orders O and I *O are isomorphic for any locally free left O-module I of rank 1.

Lemma 4.7: Let M be a (O, S)-bimodule and let I be a locally free left-O-module of rank 1. Then I-1 <o M admits a structure of (I * O, S)-bimodule and the isomorphism O = I *O identifies the (O, S)-bimodule M with the (I * O, S)-bimodule I-1 <O M.

Proof. Since B is indefinite, Pic(^, N) = 1 and I must be principal. Write I = Oj. Then I *O = y -1oy and the isomorphism I *O = O is given by

Y 1aY — a. Finally, the isomorphism of Z-modules

I-1 ®a M —> M

Y-13 ® m -—> ¡m

is compatible with the isomorphism I * O = O described above. I

Theorem 4.8: The map $ : CMD,N(R) —> |J M CMMS(R) satisfies the reciprocity law

$([J] * p) = [J]-1 * $(p), for any p : R—O in CMd,n (R) and any [J] e Pic(R). Proof. The map $ is given by

$ : CMd,n(R) ^ Um CMMMS(R)

(p : R—O) (O0R S ,$v : R — EndO(O<g>R S)).

Let p : R—O denote the conjugacy class of an embedding in CMD,N(R) and let [J] e Pic(R). Write [J] * p : R— [J] *O for the embedding induced by the action of [J] on p. By Lemma 4.5,

([J] *O) ®R S = (p(J-1)O Op(J)) ®R S.

Then Lemma 4.7 asserts that, under the isomorphism [J] * O = O, the ([J] * O, S)-bimodule (p(J-1 )O ®q Op(J)) ®R S is naturally identified with the (O, S)-bimodule Op(J) ®R S. Moreover, the embedding $y]*v is given by

R — EndO (Op(J) ®r S)

S -—> (ap(j) ® s — ap(jS) ® s).

Setting A = EndS (O ®R S), we easily obtain that

Op(J) ®r S = $(p)(J)A (O®r S) = [J]-1 * (O®r S).

Finally, since the action of $j]*V(R) on [J]-1 * (O®R S) is given by the natural action of R on J, we conclude that <f>([J] * p) = [J]-1 * 4>(p)- ■

Let M be an admissible (O, S)-bimodule of rank 8. By Theorem 3.7, the map M -—> Endf, (M) induces a one-to-one correspondence between the sets PicO(M) and Pic(DM,NNM), where DM and NM were already defined in §3. This implies that the set CMM5(R) can be identified with CMdm nmn(R), under the above correspondence.

Both sets are endowed with an action of the group Pic(R). We claim that the bijection

(4.9) CMmms(R) ^ CMdmnmn(R)

is equivariant under this action.

Indeed for any [J] G Pic(R) and any (N,— : R H End| (N)) in CMMs(R), Theorem 4.6 asserts that HomO(N, [J] *N) = —(J-1)EndO(N). Therefore

EndO([J] *N) ={p G Endf(N0) : pHomSO(N, [J] *N) C HomO(N, [J] *N)} = [J] * EndS (N),

which is the left Eichler order of —(J-1)EndS(N). Moreover, [J] *— arises from the action of R on —(J -1)EndS (N) via —. In conclusion, both actions coincide.

Corollary 4.9: Assume that for all P G CM(R) the bimodules Mp are admissible. Then the map $ : CMD,N(R)^LIm CMdm nnm (R) satisfies $([J]*p) = [J]-1 *$(p), for any p : R H O in CMD,N(R) and any [J] G Pic(R).

4.2. Atkin-Lehner involutions. Recall from §2.1 that the set of optimal embeddings CMD,N(R) is also equipped with an action of the group W(D,N) of Atkin-Lehner involutions. Let q | DN, q = p be a prime and let n > 1 be such that qn || DN. For any (O, S)-bimodule M, there is a natural action of wqn on CMM5(R) as well, as we now describe.

Let Qs be the single two-sided O-ideal of norm qn. Let

(N: Rh EndS (N)) G CMMMS(R)-

Since Qs is two-sided, Qs®sN acquires a natural structure of (O, S)-bimodule.

Recall that O equals Qs*O as orders in B. Hence the algebra EndS(Qs<8>sN) is isomorphic to EndQQ0^s(Qs ®S N) = EndS(N) by Lemma 4.7.

Moreover, the bimodule Qs ®s N is locally isomorphic to N at all places of Q except possibly for q. Since we assumed q = p, there is a single isomorphism class locally at q. Hence we deduce that Qs ®s N G Pic^ (M). Thus wqn defines an involution on CMM5(R) by the rule wqn (N, —) = (Qs ®S N, —).

We now proceed to describe the behavior of the Atkin-Lehner involution wqn under the map

$ :CMd,n(RHU CMo!s(R)

introduced above.

Theorem 4.10: For all p : R — O in CMD,N(R),

$(wqn (p)) = wqn ($(p)).

Proof. For any p e CMD,N (R), let wqn (p) : R — QO *O as in §2.1. Set $(p) = (O®r S, $v).

As ring monomorphisms, p equals wqn (p). Hence,

$(Wqn (p)) = ((Qo *O) ®r S,$v ).

Applying Lemma 4.5, we obtain that

(Qo *O) ®R S = (QO1 ®o Qo) ®R S.

By Lemma 4.7, the (Qo * O, S)-bimodule (Q—1 Qo) &>R S) corresponds to the (O, S)-bimodule Qo ®R S = Qo ®o (O ®R S). Thus we conclude that

4>{wqn{tp)) = (Qo (o S), 4>v) = wqn(4>(tp)). I

Remark 4.11: We defined an action of wqn on CMMS(R) for any (O,S)-bimodule M. Returning to the situation where M is admissible, one can ask if, through the correspondence CMMS(R) ^ CMdm nmn(R) of (4.9), this action agrees with the Atkin-Lehner action wqn on CMdm nmn(R).

Indeed, let (N,^) e CMMS(R) and set A = EndO(N) e Pic(DM,N0MN). Since wqn (N, = (Qo ®o N,4>), we only have to check that Qa := HomS (N, Qo ®o N) is a locally free rank-1 two-sided A-module of norm qn in order to ensure that EndS(Qo ®o N) = Qa * A.

Since Qa is naturally a right EndS (Qo ®o N)-module and A equals EndS (Qo ®o N) as orders in End^ (N0), we conclude that Qa is two-sided. In order to check that Qa has norm qn, note that Qa coincides with HomO(Qo ®O N, (Qo ®O QO) ®O N) as ideals on the ring A. Hence

QA = HomO (Qo ®o N, qnN)-HomO (N, Qo ®o N) = HomO (N, qnN) = qnA.

5. CCerednik—Drinfeld's special fiber

In this section we exploit the results of §4 to describe the specialization of Heegner points on Shimura curves X0(D, N) at primes p | D. In order to do so, we first recall basic facts about the moduli interpretation of Cerednik-Drinfeld's special fiber of X0(D, N) at p.

Let p be a prime dividing D, fix F an algebraic closure of Fp and let X0(D, N)= X0(D,N) x Spec(Fp). By the work of Cerednik and Drinfeld, all irreducible

components of X0(D,N) are reduced smooth conics, meeting transversally at double ordinary points.

Points in X0(D,N)(F) parameterize abelian surfaces (A,i) over F with QM by O such that TraceF(i(a) | Lie(A)) = Tr(a) G Q for all a G O. Here Tr stands for the reduced trace on O.

It follows from [23, Lemma 4.1] that any abelian surface with QM by O in characteristic p | D is supersingular. Moreover, singular points of X0(D,N) correspond to abelian surfaces (A, i) with QM by O such that a(A) = 2 and their corresponding bimodule is admissible at p, of type (1,1) (cf. [23, §4]). Observe that p is the single place at which both O and S ramify. Hence for such bimodules we have S = {p}, D0 = D/p and N0M = p in the notation of §3.1.

Remark 5.1: In Ribet's original paper [23], singular points are characterized as triples [A0,i0,C], where (A0,i0) is an abelian surface with QM by a maximal order and C is a r0(N)-structure. According to Appendix A, we can construct from such a triple a pair (A, i) with QM by O such that End(A, i) = End(A0,i0,C).

Let P = [A, i] G CM(R) be a Heegner point and let G CMDjW(R) be the optimal embedding attached to P. It is well-known that if such an optimal embedding exists, R is maximal at p and p either ramifies or is inert in K (cf. [27, Theorème 3.1]). Thus, [A, i] is supersingular in X0(D,N) and the bimodule associated to P is Mp = O S, by Theorem 4.3. Moreover, by Remark 4.2 the embedding is optimal.

Proposition 5.2: Assume that p ramifies in K. Then Mp is admissible at p and (Mp )p is of type (1,1). Furthermore, the algebra End^ (Mp ) admits a natural structure of oriented Eichler order of level Np in the quaternion algebra

of discriminant —.

Proof. Since Op = Sp is free as right Rp-module, we may choose a basis of Op over Rp. In terms of this basis, the action of Op on itself given by right multiplication is described by a homomorphism

f : Op ^ M2(Rp).

Since p ramifies in K, the maximal ideal pop of Op is generated by an uni-formizer n of Rp (cf. [27, Corollaire II.1.7]). This shows that f (pOp) Ç M2(nRp).

This allows us to conclude that Mp is admissible at p. Indeed, in matrix terms, the local bimodule Mp is given by the composition of f with the natural inclusion M2(Rp) M2(Sp). Notice that M2(nRp) is mapped into M2(psp) under that inclusion.

In order to check that rp = 1, reduction modulo pop yields an embedding

f : O/po ^ M2(Fp)

because the residue field of K at p is the prime field Fp. After extending to a quadratic extension of Fp, this representation of O/po = Fp2 necessarily splits into the direct sum of the two embeddings of Fp2 into F. On the other hand, we know that Mp = Op x pOp with s + r = 2. Consequently, we deduce that Mp = Op x pOp and Mp is of type (1,1).

Finally, since in this case Do = y and No = p, it follows from Theorem 3.7 that End%{M) G £{D/p, Np) . I

Theorem 5.3: A Heegner point P e CM(R) of X0(D, N) reduces to a singular point of X0(D, N) if and only if p ramiGes in K.

Proof. As remarked above, p is inert or ramifies in K. Assume first that p ramifies in K. Then by Proposition 5.2 the bimodule Mp is admissible at p and of type (1,1). It follows from [23, Theorem 4.15, Theorem 5.3] that the point P e X0(D, N) is singular.

Suppose now that P = [A, i] is singular. Then its corresponding (O, S)-bimodule Mp is admissible at p and of type (1,1). Thus End(A, i) = EndO (Mp) e Pic(D/p, Np) and the conjugacy class of the optimal embedding p : R ^ Endf, (M p) is an element of CMb Np(R). If such an embedding exists, then p cannot be inert in K thanks to [27, Theoreme 3.2]. Hence p ramifies in K. I

That points P e CM(R) specialize to the non-singular locus of X0(D,N) when p is unramified in K was already known by the experts (cf., e.g., [15]) and can also be easily deduced by rigid analytic methods.

5.1. Heegner points and the singular locus. As explained before, singular points of X0(D,N) are in one-to-one correspondence with isomorphism classes of (O, S )-bimodules which are admissible at p and of type (1,1). By Theorem 3.7, such an isomorphism class is characterized by the isomorphism class of the endomorphism ring Endf, (M), which is an oriented Eichler order of level

Np in the quaternion algebra of discriminant D/p. Thus the set X0(D,N)s¡ng of singular points of X0(D,N) is in natural one-to-one correspondence with Pic(D/p,Np).

Let P = [A,i] G CM(R) be a Heegner point. As proved in Theorem 5.3, P specializes to a singular point P if and only if p ramifies in K. If this is the case, the optimal embedding $P : R ^ Endf, (Mp) provides an element of CM m„p(R).

Therefore, the map $ of (4.7), which was constructed by means of the reduction of Xo(D, N) modulo p, can be interpreted as a map between CM-sets:

(5.10) $ : CMD,N(R) —► CM.a ^(fi).

Moreover, composing with the isomorphism CM(R) ~ CM^,w(R) of (2.4) and the projection tt : CMb NJR)^Pic(^,Np) of (4.8), the resulting map

-p i P P

CM(i?)—»Pic(y, Np) describes the specialization of P G CM(fi) at p. Theorem 5.4: The map $ : CMd,n (R) —>• CMb

,Np(R) is equivariant for the action of W(D,N) and, up to sign, of Pic(R). More precisely:

$([J] * y) = [J]-1 * $(wm(p)) = Wm ($(p))

for all m II DN, [J] G Pic(R) and y : R^O in CM D,N (R). Moreover, if N is square-free, $ is bijective.

Proof. The statement for Pic(R) is Corollary 4.9. It follows from Theorem 4.10 and Remark 4.11 that $(wm(y)) = wm($(y)) for all m || ND/p. Since p ramifies in K, wp G W(D,N) = W(^,Np) preserves local equivalence by [27, Theoreme II.3.1]. Hence the action of wp coincides with the action of some [P] G Pic(R). This shows that $(wm(y)) = wm($(y)) for all m || ND.

Finally, in order to check that $ is a bijection when N is square-free, observe that Pic(fi) x Wd,n{R) = Pic(R) x Wd Np(R) acts freely and transitively both on CMDiN(R) and on GMo Np(R). " I

5.2. Heegner points and the smooth locus. In [23, §4], Ribet describes the smooth locus X0(D,N)ns of X0(D,N) in terms of abelian surfaces (A, i) with QM over F. We proceed to summarize the most important ideas of such a description.

Let (A, i) be an abelian variety with QM by O. By [23, Propositions 4.4 and 4.5], the isomorphism class [A, i] defines a non-singular point P G X0(D, N)ns if

and only if A has exactly one subgroup scheme Hp, which is O-stable and isomorphic to ap. There ap stands for the usual inseparable group scheme of rank p. Furthermore, by considering the quotient B = A/Hp and the embedding j : O — EndFB induced by i, we obtain an abelian surface (B,j) with QM such that a(B) = 2. The pair (B,j) defines an admissible bimodule MP = M(p ^ which has either type (2, 0) or type (0, 2).

The set of irreducible components of X0(D,N) is in one-to-one correspondence with the set of isomorphism classes of admissible (O, S)-bimodules of type (2,0) and (0, 2). With this in mind, the bimodule MP determines the component where the point P lies. Moreover, the Atkin-Lehner involution wp e EndQ(X0(D, N)) maps bimodules of type (2,0) to those of type (0, 2), and vice versa. By Theorem 3.7, such bimodules M(_p j are characterized by their type and their endomorphism ring = End(B,j) G Pic(y,Ar).

Hence the set of irreducible components of X0(D,N) is in one-to-one correspondence with two copies of Picone copy for each type, (0,2) and

(2, 0). Finally, the automorphism wp exchanges both copies of Pic(y, N).

Let P = [A, i e X0(D, N) be a non-singular point and assume that a(A) = 2. Let Mp be its associated (O, S)-bimodule. The subgroup scheme Hp gives rise to a degree-p isogeny up : A—B = A/Hp such that upi(a) = j(a)up for all a eO.

Since A ~ B ~ E2, we may fix an isomorphism of algebras EndF(A)—M2(S). Then each isogeny can be regarded as a matrix with coefficients in S. In order to characterize the bimodule MP in terms of Mp we shall use the following proposition.

Proposition 5.5: Let (A, i)/F and (A, j)/F be abelian surfaces with QM by O such that a(A) = 2. Let M and M be their associated (O, S)-bimodules. Consider the usual morphisms attached to each bimodule

fM : O-^M2(S), fft : O-^M2(S),

and assume there exists an isogeny y : A —> A such that fM(a)Y = Yfj^(a) for all a eO. Then the image y M, where M is viewed as a free right S-module of rank 2, is O-stable. Furthermore, M = yM as (O, S)-bimodules.

Proof. The right S-module yM is S-free of rank 2 with basis {Yei,Ye2}, provided that {e1,e2} is a S-basis for M. An element of yM can be writen as

Yeia + ie2b = ^ Yei 7e2 ) ^ b ^ = ( ei e0 Y

Therefore, any a eO acts on it as follows:

( ei e2 ) fM(a)Y

= ei e2

jei Ye2

) YfM*)( a)

Thus yM is O-stable and O acts on it through the map fft. We conclude that 7M = M as (O, <S)-bimodules. I

Applying the above proposition to fM = j, fft = i and y = lp, we obtain that |pMp = Mp.

Note that endomorphisms A e End(A) which fix Hp give rise to endomor-phisms A e End(B). If in addition A e End(A, i), then A lies in End(B, j).

Lemma 5.6: Every endomorphism in End(^4, i) fixes H

Proof. Let A e End(A, i). Then A(Hp) is either trivial or a subgroup scheme of rank p. If A(Hp) = 0, the statement follows. Assume thus that A(Hp) is a subgroup scheme of rank p. Since A is supersingular, A(Hp) is isomorphic to ap. Moreover, for all a eO we have that i(a)(A(Hp)) = A(i(a)(Hp)) = A(Hp). Therefore A (Hp) is O-stable and consequently A (Hp) = Hp, by uniqueness. I

By the preceding result, there is a monomorphism End(A, i)—End(B, j) corresponding, via bimodules, to the monomorphism dp :End^(Mp)—End^(MP) that maps every A e Endf,(Mp) to its single extension to MP D Mp.

Let (N,^) e CM^p1 (R). By §3.2, N = Mp p,) for some abelian surface (A', i') with QM by O. By [23, Theorem 5.3], whether [A', i'\ is a singular point of X0(D,N) or not depends on the local isomorphism class of M(p,p,) at p.

and Mp are locally isomorphic, [A',i'] = Q e X0(D,N)ns and

Since M

(A',i')

we may write M(j, j,) = Mj. The composition of ^ with Sq gives rise to an embedding R Endf,(MQ), which we claim is optimal.

Indeed, since /i,q has degree p, [MQ : Mq] = p. Hence the inclusion EndO(Mq) C EndO(MQ) has also p-power index. Due to the fact that 4 is optimal,

Sq o 4(R) = Sq(EndO(Mq) n 4(K)) Ç EndO(MQ) n Sq o ) =: Sq o 4(R'),

where R' is an order in K such that R Ç R' has p-power index. Recall that R is maximal at p, hence we conclude that R' = R and Sq o 4 is optimal.

Since Hq is the single O-stable subgroup scheme of A' of rank p, MQ is the single O-stable extension of Mq of index p. Moreover, it is characterized by the local isomorphism class of MQ at p. More precisely, if Mq and Mq are locally isomorphic, then MP and MQ must be locally isomorphic also. Thus the correspondence (N = Mq, 4) ^ (MQ, Sq o 4) induces a map

(5.11) S : CM^JQ (R)^ CMM (R).

Both sets are equipped with an action of Pic(R) and involutions wqn, for all qn || DN, q = p. On the other hand, since MP is admissible of type (2,0) or (0, 2), we identify CM^ (R) with CMD/pN(R) by Theorem 3.7 .

Lemma 5.7: The map S : CM^p(R)^ CMD/p N(R) is equivariant under the actions of Pic(R) and W(D/p, N).

Proof. Let (Mq,4) G CM;Mf (R), let [J] G Pic(R) and write A = EndO(Mq). We denote by Q[J] e X0(D, N) the point attached to the bimodule [J] * Mq = 4(J -1)A Mq = Mqj .

Write A' = EndO (MQ ) D A and consider the bimodule [J] * MQ = 4(J-1)A' <g>A, MQ G PicO(MP). Then [J] * Mq C [J] * MQ is O-stable of index p and [J] * MQ = MQl 1 by uniqueness. Since [J] * 4 and [J] * (Sq o 4) are given by the action of R on Mq[j1 and MQlJ] respectively,

S([J]*(Mq, 4)) = (MQ[J1, Sq[J] O[J]*4) = ([J]*MQ, [J]*(Sqo^))= [J]*S(mq, 4).

As for the Atkin-Lehner involution wqn, let Qo be the single two-sided ideal of O of norm qn. Again Qo ®o MQ is an O-stable extension of Qo ®o Mq of index p. Hence QO ®O MQp = Mwqn (¿P), where wqn (Q) e X0(D,N) is the non-singular point attached to Qo ®o Mq. This concludes that S o wqn =

wqn O Ô. I

Let P = [A,i] e CM(R) be a Heegner point such that P e X0(D,N)ns. Let Mp be the bimodule attached to its specialization P = [A, i] e X0(D, N). Recall the map $ : CMD,NRH LIM CMMsS(R) of (4.7), induced by the natural injection End(A, i) ^ End(A, i) and the correspondences of (2.4) and (3.5). The aim of the rest of this section is to modify $ in order to obtain a map

$ : CMd,n(R)^ CMb n(R) U CM 2. n(R)

p ' p '

which, composed with the natural projection

tr : CM£ n(R) UCMz, n(R)—Picf —,N) UPicf — ,N

P' P' \P J \P

and the correspondence of (2.4), assigns to P e CM(R) the Eichler order that describes the irreducible component at which P specializes.

Theorem 5.8: The embedding End(A,i)m-End(Aj)m-End(B, j)ePic(f,N) induces a map

(5.12) $ : CMD n(R)— CM b N(R) U CM£ N(R),

' p ' p '

which is equivariant for the action ofWo_ N(R) and satisfies the reciprocity law

$([J] * p) = [J]-1 * $(p) for all [J] e Pic(R) and p e CMD,N(R).

If N is square-free, then $ is bijective, i.e., it establishes a bijection of Wd N(R)-sets between CMD N(R)/wp and CMb at(R).

p ' ' p'

Notice that we are considering N) as a subgroup of W(D, N), so that

acts naturally on CMd,n(R)- Since points in CM(i?) have good reduction, p is inert in K by Theorem 5.3. Therefore, by [27, Theoreme 3.1],

rrip = 2 and the subgroups W]j n (R)/(wp) and Wd n(R) are isomorphic. Hence

we can consider CMjj n(R)/wp as a Wn N(R)-set.

Proof The map CMD n(R)—CM d n(R) U CM d n(R) arises as the com' p ' p '

position

CMdn(R) A y CMMS(R) a CMd/pn(R) U CMd/pn(R),

where S takes, via the map denoted by the same symbol in Lemma 5.7, an element of CMMsS(R) to the first component of CMD/pN (R) U CMD/pN (R) if, say, M is of type (2, 0) and to the second component otherwise. Then Lemma 5.7, Theorem 4.8 and Theorem 4.10 prove the first statement.

Assume that N is square-free. Then freeness and transitivity of the action of Wd/p n(R) x Pic(R) on both CM^ n(R)/wp and CMb m(R) show that the corresponding map between them is bijective. Since the Atkin-Lehner involution wp exchanges the two components of CMD/pN (R) U CM d/p,n (R), we conclude that $ : CMDiN(R)^CMD/PtN(R) U CMd/p,n(R) is also bijective. I

6. Supersingular good reduction

Exploiting the results of §4, we can also describe the supersingular reduction of Heegner points at primes p of good reduction of the Shimura curve X0(D, N). Indeed, let p \ DN be a prime, let F be an algebraic closure of Fp and let X0(D, N) = X0(D, N) x Fp. If P = [A,!] e X0(D, N) is a supersingular abelian surface with QM, then a(A) = 2 by [23, §3]. Therefore, [A, i] is characterized by the isomorphism class of the (O, S)-bimodule Mp attached to it. By Theorem 3.7, these isomorphism classes are in correspondence with the set Pic(Dp, N).

Let P = [A, i] e CM(R) be a Heegner point and assume that the conductor c of R is coprime to p. Since A is isomorphic to a product of two elliptic curves with CM by R, it follows from the classical work of Deuring that A has supersingular specialization if and only if p does not split in K. If we are in this case, the map $ of (4.7) becomes

(6.13) $ : CMd,N(RH CMDP,N(R)

by means of the natural identification (4.9). Composing with the natural projection

n : CMdp,n(R)HPic(Dp,N)

and the correspondence of (2.4), one obtains a map CM(R) H Pic(Dp, N) which assigns to P = [A, i] e CM(R) the Eichler order End(A, i) that describes its supersingular specialization.

Theorem 6.1: The map $ : CMD,N(R) —> CMDp,N(R) is equivariant for the action of W(D,N) and, up to sign, of Pic(R). More precisely

$([J] * y) = [J]-1 * $(y), $(wm(y)) = Wm ($(y))

for all m II DN, [J] e Pic(R) and y : RhO in CMD,N(R).

Assume that N is square-free. If p ramifies in K, the map $ is bijective. If p is inert in K, the induced map

CMD ,n(R) H CMdp,n(R)h CMdp,n (R)/wp

is also bijective.

Proof. The first statement follows directly from Theorem 4.8 and Theorem 4.10. Assume that N is square-free. If p ramifies in K, we have that WD,N (R) = WDp N (R). Then Pic(R) x WD,N(R) acts simply and transitively on both CMD N (R) and CMD,N(R) and $ is bijective. If p is inert in K, then WD,N(R)= W]Jp,n(R)/wp and the last assertion holds. I

7. Deligne—Rapoport's special fiber

In this section we exploit the results of §6 to describe the specialization of Heegner points on Shimura curves X0(D, N) at primes p || N. In order to do so, we first recall basic facts about Deligne-Rapoport's special fiber of X0(D, N) at p.

Let p be a prime dividing exactly N, fix F an algebraic closure of Fp and let X0(D,N) = X0(D,N) x Spec(Fp). By the work of Deligne and Rapoport, there are two irreducible components of X0(D, N) and they are isomorphic to X0(D, N/p). Notice that X0(D, N/p) has good reduction at p, hence X0(D, N/p) is smooth.

Let X0(D,N) ^ X0(D, N/p) be the two degeneracy maps described in Appendix A. They specialize to maps X0(D,N) ^ X0(D,N/p) on the special fibers; we denote them by 5 and 5wp. Write 7 and wp7 for the maps X0(D, N/p) ^ X0(D, N) given in terms of the moduli interpretation of these curves described in Appendix A by 7((A, i)) = (A, i, ker(F)) and wpy((A, i)) = (A(p),i(p), ker(V)), where F and V are the usual Frobenius and Verschiebung. It can be easily checked that 7 o 5 = wp7 o 5wp = Id and 7 o 5wp = wp7 o 5 = wp.

According to [3, Theorem 1.16], 7 and wpy are closed morphisms and their images are respectively the two irreducible components of X0(D, N). These irreducible components meet transversally at the supersingular points of X0(D, N/p). More precisely, the set of singular points of X0(D, N) is in one-to-one correspondence with supersingular points of X0(D, N/p).

Let P = [A, i] e CM(R) be a Heegner point and let P = [A,T] e X0(D, N) denote its specialization. According to the above description of the special fiber,

the point P is singular if and only if S(A,i) is supersingular, or, equivalently, if and only if (A, i) is supersingular because (A, i) and 6(A, i) are isogenous. By [27, Theoreme 3.2], the fact that CM(R) = 0 implies that p is not inert in K. Since A is the product of two elliptic curves with CM by R, we obtain the following result, which the reader should compare with Theorem 5.3.

Proposition 7.1: A Heegner point P e CM(R) reduces to a singular point of X0(D, N) if and only if p ramiGes in K.

Proof. The point P specializes to a singular point if and only if (A, i) is supersingular. Since p is not inert in K and any elliptic curve with CM by R has supersingular specialization if and only if p is not split in K, we conclude that p ramifies in if. I

Remark 7.2: In this section we only consider the case when p || N does not divide the conductor of R. The case p dividing the conductor of R is also important for applications to Iwasawa theory. In [1, Lemma 3.1] a similar result is proved: If R C K is an order of conductor pnc and p is inert in K, then any Heegner point P e CM(R) specializes to a singular point of X0(D, N).

7.1. Heegner points and the smooth locus. Let O' D O be an Eichler order in B of level N/p. Notice that O' defines the Shimura curve X0(D, N/p). Let CM(R) be a set of Heegner points that specialize to non-singular points in X0(D,N) and let P e CM(R). The inclusion O' DO defines one of the two degeneracy maps d : X0(D,N) — X0(D, N/p) as in Appendix A. By the identification of (2.4), P corresponds to pP e CMD , N(R) and its image d(P) corresponds to pd(p) e CMD ,N/p(R'), where pP(R') = pP(K) n O' and pd(p) : R' — O' is the restriction of pP to R'. Since [O' : O] = p, the inclusion R C R' has also p-power index. According to the fact that the conductor of R is prime to p, we deduce that R = R'.

Hence, restricting the natural degeneracy maps X0(D, N) ^ X0(D, N/p) to CM(R), we obtain a map

(7.14) CMD , n(R)— CMD , n/p(R) u CMD , N/p(R).

Observe that we have the analogous situation to §5.2 and Theorem 5.8. We have a map CMD,N(R)— CMD ,N/p(R) U CMD ,N/p(R), which is clearly Pic(R) x W(D, N/p) equivariant and a bijection if N is square-free, with the

property that the natural map

CMd,n/p(R) U CMAN/p(R)HPic(D, N/p) U Pic(D, N/p)

gives the irreducible component where the point lies. Notice that there are two irreducible components and #Pic(D, N/p) = 1, since D is the reduced discriminant of an indefinite quaternion algebra.

7.2. Heegner points and the singular locus. Let O' D O be as above, let (A, i) be an abelian surface with QM by O' and let C be a r0(p)-structure. Given the triple (A,i,C), write P = [A,i,C] for the isomorphism class of (A, i, C), often regarded as a point on X0(D, N) by Appendix A.

Let P = [A, i, C] e CM(R) be a Heegner point with singular specialization in X0(D, N). Then [A, i] e X0(D, N/p) is the image of P through the natural map d : X0(D,N)—yX0(D, N/p) given by O' D O. Using the same argumentation as in the above setting, we can deduce that End(A, i) = End(A, i, C) = R.

Let [A, i, C] e X0(D,N) be its specialization. Since it is supersingular, C = ker(Fr). Thanks to the fact that the Fr lies in the center of End(A), we obtain that End(A,i,C) = End(A, i). Thus the embedding End(A, i, C) — End(A, i, C) is optimal and it is identified with End(A, i) — End(^4, i), which has been considered in §6. In conclusion, we obtain a map $ : CMD,N(R) —> CMDp N/p(R) as in §5.1 and, since p ramifies in K, a result analogous to Theorem 5.4.

Theorem 7.3: The map

(7.15) $ : CMDiN{R) ^ CMDpx{R)

is equivariant for the action of W(D,N) and, up to sign, of Pic(R). More precisely

$([J] * y) = [J]-1 * $(y), $(wm(y)) = wm ($(y))

for all m II DN, [J] e Pic(R) and y : R — O in CMDN(R). Moreover, it is bijective if N is square-free.

8. Numerical example

Let X = X0(77,1) be the Shimura curve of discriminant D = 77. By the theory of Cerednik-Drinfeld, the dual graph of the special fiber X of X at p = 7 is

given by (cf. [13, §3] for a step-by-step guide on the computation of these graphs, using MAGMA)

¿(v2) = 3

£(vi) = 2

l{v'2) = 3 1 ¿(>i) = 2

Hence the special fiber X has the form

where each of the four irreducible components v2,v2,v3,v'3 are smooth curves of genus 0 over F7, meeting transversally at points whose coordinates lie in F49. The involution w11 fixes each of these components, while the straight horizontal line is the symmetry axis of the Atkin-Lehner involution w7. Let x1 ,y1,x3,y3 denote the four intersection points which lie on this axis. The subindex 1 or 3 stands for the thickness of the points (cf. § 1 for the definition).

Let K be the imaginary quadratic field Q(a/—77) and let R be its maximal order. By Theorem 5.3, any Heegner point in CM(R) specializes to a singular point. By [20, §1], every P e CM(R) is fixed by w77 and it follows from the above discussion that its specialization at XX lies in the symmetry line of w7. Thus P is one of the singular points {x1, yi.,X3, y3}.

By Theorem 5.4 and upon the identification of (2.4), the map $ provides a bijection between CM(R) and CM117(R). Moreover, composing with the natural projection n : CM117(R)—Pic(11, 7), one obtains the specialization of the points in CM(R) at X. Via $, the action of Pic(R) on CM117(R) becomes the Galois action of Gal(HR/K) ^ Pic(R) by Shimura's reciprocity law [24, Main Theorem I]. This allows us to explicitly compute the specializations of the points P e CM(R) to the set {x1,y1,x3,y3}.

Namely, we computed with Magma that Pic(R) = ([/], [J]) ~ Z/2Z x Z/4Z. Let

a : Pic(R)—Gal(Ffl/K) stand for the inverse of Artin's reciprocity map. We obtained that P and Pj2I}) specialize to the same singular point of thickness 3, say x3; P^(W) and Pa([jiI}) specialize to the same singular point of thickness 1, say x1; Pct([j]) and Pa([j3]) specialize to y1; and Pa([j2]) and Pa([JI} specialize to y3.

Appendix A: Moduli interpretations of Shimura curves

In this appendix, we describe an interpretation of the moduli problem solved by the Shimura curve X0(D,N), which slightly differs from the one already considered in §2.2. This moduli interpretation is also well-known to the experts, but we provide here some details because of the lack of a suitable reference.

Let {ON}(n,d)=1 be a system of Eichler orders in B such that each ON has level N and ON C OM for M | N. Let now M || N, by which we mean a divisor M of N such that (M, N/M) = 1. Since ON C OM, there is a natural map S : X0(D,N) — X0(D,M); composing with the Atkin-Lehner involution wN/M, we obtain a second map S o wN/M : X0(D,N) — X0(D,M) and the product of both yields an embedding j : X0(D, N) — X0(D, M) x X0(D, M) (cf. [9] for more details).

Note that the image by j of an abelian surface (A, i) over a field K with QM by ON is a pair ((Ao,i0)/K, (A0,i0)/K) of abelian surfaces with QM by OM, related by an isogeny $n/m : (A0,i0)—(A0,i0) of degree (N/M)2, compatible with the multiplication by OM. Assume that either char(K) = 0 or (N, char(K)) = 1, thus giving a pair ((A0,i0), (A0,i0)) is equivalent to giving the triple (A0,i0,CN/M), where CN/M = ker($n/m) is a subgroup scheme of A0 of rank (N/M)2, stable by the action of OM and cyclic as an OM-module. The group CN/M is what we call a r0(N/M)-structure.

Let us explain now how to recover the pair (A, i) from a triple (A0, i0, CN/M) as above. Along the way, we shall also relate the endomorphism algebra End(A0,i0,CN/M) of the triple to the endomorphism algebra End(A, i). The construction of the abelian surface (A,i) will be such that the triple (A0 ,i0,CN/M) is the image of (A, i) by the map j. This will establish an equivalence of the moduli functors under consideration, and will allow us to regard points on the Shimura curve X0(D, N) either as isomorphism classes of abelian surfaces (A, i)

with QM by ON or as isomorphism classes of triples (A0,i0,CN/M) with QM by OM and a r0(N/M)-structure, for any M || N.

Since CN/M is cyclic as a OM-module, CN/M = OMP for some point P e A0. We define

Ann(P) = {I e Om : IP = 0}.

It is clear that Ann(P) is a left ideal of OM of norm N/M, since CN/M = OM/Ann(P). Let a be its generator. Assume C = CN/M n ker(i0(a)) and let A := A0/C be the quotient abelian surface, related with A0 by the isogeny y : A0—A. We identify

C = {pP : I e Om , ap e Ann(P) = Om a} = {IP : I e a-1OM a n Om }■

The order a-1OM a n OM is an Eichler order of discriminant N, hence C = {IP, s.t. | e On}.

Since ON = {I e OM : IC C C} C OM, the embedding i0 induces a monomorphism i : ON — End(A) such that i(B) n End(A) = i(ON). Then we conclude that (A, i) has QM by ON. Recovering the moduli interpretation of X0(D, N), it can be checked that 5(A, i) = (A0, i0).

Let End(A0, i0, CN/M) be the subalgebra of endomorphisms ^ e End(A0, i0), such that

V>(Cn/ M ) C CN/M ■

Identifying End(A0,i'0) inside End0(A0,i0) via $N/M, we obtain that

End(A0, i0, Cn/m) = End(A0, i0) n End(A0, i0).

Let ^ e End(A0,i0,CN/M). Since ^ commutes with i0(a), we have that -0(ker(i0(a)))Cker(i0(a)). Moreover, it fixes CN/M by definition, hence ^(C)CC and therefore each element of End(A0, i0,CN/M) induces an endomorphism of End(A,i). We have obtained a monomorphism End(A0,i0,CN/M) — End(A,i).

Proposition 8.1: Let (A,i) be obtained from (A0,i0,CM/N) by the above construction. Then End(A, i) ~ End(A0, i0, CN/M).

Proof. We have proved that End(A0, i0, CN/M) — End(A, i); if we check that End(A, i) — End(A0,i0) and End(A, i) — End(A0,i0), we will obtain the equality, since End(A0, i0, CN/M) = End(A0, i0) n End(A0, i0).

Due to the fact that C C CN/M, the isogeny &N/M factors through p. By the same reason i0(a) also factors through p, and we have the following commutative diagram:

Ao^lAo

Ai> -A A

We can suppose that a is a generator of the two-sided ideal of ON of norm N/M. It can be done since ON = OM n a-1OMa, thus a differs from a chosen generator of this ideal just by a unit of OM, namely an isomorphism in Endc (A0).

The orientation ON C OM induces an homomorphism, p : ON—ON/OMa = Z/-p-Z. We consider the subgroup scheme

C1 = {P e ker(i(a)) : i(3)P = p(3)P, for all 3 e ON}. Claim: C1 = ker(n).

Clearly ker(n) C ker(i(a)). Moreover, ker(n) = p(ker(i0(a))) are those points in ker(i(a)) annihilated by i(OMa), thus they correspond to the eigenvectors with eigenvalues p(3), for all 3 eON.

We also have the homomorphism, ¡j o w^/m '■ On—On/uOm — Again, we consider the subgroup scheme

C2 = {P e ker(i(a)) : i(3)P = p o wn/m(3)P, for all 3 e On}■ Claim: C2 = ker(p).

In this case, we have that ker(p) = p(CN/M), where CN/M = OMP. Due to the fact that aOM C ON and ker(p) = ONP, we obtain that ker(p) C ker(i(a)). By the same reason, ker(p) is the subgroup of ker(i(a)) annihilated by i(aOM), therefore ker(p) = C2 as stated.

Finally, let 7 e End(A, i). Since it commutes with i(3) for all 3 e ON, it is clear that 7(C2) C C2. Then the isogeny p induces an embedding End(A, i) C End(Ao,ig). Furthermore, we have that io(a)io(a) = N/M, then p = rj o i0(a) = i(a) o r/. Since 7 commutes with i(a) and 7(C*i) C Ci, we obtain that p induces an embedding End(A, i) C End(A0,i0). We conclude that End(A, i) = End(A0, i0) D End(A^, i'0) = End(A0, i0, CN/M). I

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