SINGULAR MODULI REFINED

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1 SINGULAR MODULI REFINED BENJAMIN HOWARD AND TONGHAI YANG 1. Introduction Let K 1 and K 2 be nonisomorphic quadratic imaginary fields with discriminants d 1 and d 2, respectively, and set K = Q( d 1, d 2 ). Let F be the real quadratic subfield of K, set D = disc(f ), and let D O F be the different of F/Q. Let x x denote complex conjugation on K and set w i = O K i. Let χ be the quadratic Hecke character of F associated to K, and let σ 1 and σ 2 be the two real embeddings of F. Throughout the introduction we assume that gcd(d 1, d 2 ) = 1. In particular this implies that K/F is unramified. Almost one hundred years ago, to construct a holomorphic modular form of (parallel) weight 1 for SL 2 (O F ), Hecke constructed the following famous Eisenstein series E (τ 1, τ 2, s) = D s+1 2 (0,0) (m,n) a 2 /O F ( π s+2 ( )) 2 s Γ 2 [a] CL(F ) (v 1 v 2 ) s 2 (m(τ 1, τ 2 ) + n) m(τ 1, τ 2 ) + n s. χ(a) N(a) 1+s Here CL(F ) is the ideal class group of F, [a] denotes the class of the fractional ideal a, and m(τ 1, τ 2 ) + n = (σ 1 (m)τ 1 + σ 1 (n))(σ 2 (m)τ 2 + σ 2 (n)). Hecke showed that this sum, convergent for Re(s) 0, has meromorphic continuation to all s and defines a (non-holomorphic) Hilbert modular form of weight 1 for SL 2 (O F ) which is holomorphic at s = 0. The value E (τ 1, τ 2, 0) at s = 0 is a holomorphic Hilbert modular form of weight 1 (Hecke s trick). He further computed the Fourier expansion of this holomorphic modular form. Unfortunately, he missed a sign in the calculation, and it turns out that E (τ 1, τ 2, 0) = 0 identically. In the early 1980 s, Gross and Zagier took advantage of this fact to compute its central derivative at s = 0, and found that the Fourier coefficients of the diagonal restriction to the upper half plane are very closely related to the factorization of singular moduli ([3]). Their result can be rephrased (see [19, Section 3] or Corollary 1.5 below for more details) in terms of arithmetic intersections as follows: if E is the moduli stack of elliptic curves over Z-schemes then the m-th Fourier coefficient of E, (τ, τ, 0) is the arithmetic intersection on E E of the m-th Hecke correspondence with the codimension two cycle of points representing pairs (E 1, E 2 ) of elliptic Date: June 1, Mathematics Subject Classification. 11G15, 11F41, 14K22. The first author is partially supported by NSF grant DMS , and by a Sloan Foundation Research Fellowship. The second author is partially supported by grants NSF DMS and NSFC

2 2 BENJAMIN HOWARD AND TONGHAI YANG curves with complex multiplication by O K1 and O K2, respectively. One naturally asks, for α F what is the arithmetic meaning of the α-th Fourier coefficient of the central derivative E, (τ 1, τ 2, 0) itself, before one restricts to the diagonal τ 1 = τ 2? In another word, is there an arithmetic Siegel-Weil formula ([8], [9]) for this Hecke Eisenstein series? The purpose of this paper is to answer this question positively. Let X be the algebraic stack over Z representing the functor which assigns to every scheme S the category X (S) of pairs (E 1, E 2 ) in which each E i = (E i, κ i ) consists of an elliptic curve E i over S and an action κ i : O Ki End(E i ). For (E 1, E 2 ) X (S) let L(E 1, E 2 ) = Hom(E 1, E 2 ) be the Z-module of homomorphisms from E 1 to E 2, equipped with the quadratic form deg. Let [, ] be the bilinear form associated to deg. The maximal order acts on L(E 1, E 2 ) by O K = O K1 Z O K2 (t 1 t 2 ) φ = κ 2 (t 2 ) φ κ 1 ( t 1 ) (t i O Ki ) making L(E 1, E 2 ) into an O K -module. The action satisfies [t φ 1, φ 2 ] = [φ 1, t φ 2 ] for all t O K, and it follows that if we view L(E 1, E 2 ) as an O F -module then there is a unique O F -bilinear form [, ] CM : L(E 1, E 2 ) L(E 1, E 2 ) D 1 satisfying [φ 1, φ 2 ] = Tr F/Q [φ 1, φ 2 ] CM. If deg CM is the O F -quadratic form on L(E 1, E 2 ) corresponding to [, ] CM then deg(φ) = Tr F/Q deg CM (φ). For any α F let X α be the algebraic stack representing the functor which assigns to a scheme S the category X α (S) of triples (E 1, E 2, j) in which (E 1, E 2 ) X (S) and φ L(E 1, E 2 ) with deg CM (φ) = α. It is clear that X α is empty unless α is totally postive. For α F totally positive define the Arakelov degree deg(x α ) = p log(p) x [X α (F alg p )] e 1 x length(o sh X α,x) where [X α (S)] is the set of isomorphism classes of objects in the category X α (S), O sh X α,x is the strictly Henselian local ring of X α at x, and e x is the order of the automorphism group of the triple (E 1, E 2, φ) corresponding to x. Define Diff( D, α) to be the set of finite primes p of F satisfying χ p (α D) = 1. If b is a fractional O F -ideal we define ρ(b) to be the number of ideals B O K satisfying N K/F (B) = b. If l is a rational prime we define ρ l (b) to be the number of ideals B O K,l satisfying N Kl /F l (B) = b l. Thus (1.1) ρ(b) = l ρ l (b). For the proof of the following theorem see Section 2.7.

3 SINGULAR MODULI REFINED 3 Theorem 1.1. If α D 1 is totally positive and Diff( D, α) = {p}, then X α has dimension zero, is supported in characteristic p = Z p, and satisfies Otherwise, one has X α =. deg(x α ) = 1 2 log(p) ord p(αpd) ρ(αdp 1 ). The functional equation forces E (τ 1, τ 2, 0) = 0, and the central derivative has a Fourier expansion E, (τ 1, τ 2, 0) = α D 1 a α (v 1, v 2 )q α where v i = Im(τ i ), e(x) = e 2πix, and q α = e(σ 1 (α)τ 1 + σ 2 (α)τ 2 ). Theorem 1.2. Suppose that α F is totally positive. If α D 1 and Diff( D, α) = {p}, then a α = a α (v 1, v 2 ) is independent of v i, and a α = 2 ord p (αpd)ρ(αdp 1 ) log p, where p is the rational prime below p. Otherwise, one has a α = 0. Theorem 1.2 is stated in a different form in [3], but without proof. We will give a sketch of the proof in Section 3 (Theorem 3.5). Combining the above theorems we obtain the following. Theorem 1.3. Assume α F is totally positive. Then X α is a stack of dimension zero and 4 deg(x α ) = a α where a α is the α-th Fourier coefficient of E, (τ 1, τ 2, 0). In Section 3 we give a slightly different and more conceptional proof of Theorem 1.3, which we now outline. Assume that α F is totally positive. In Section 2 we use Gross s work on canonical liftings of supersingular elliptic curves [2] to study the local rings OX sh α,x. These results are summarized in Theorem 3.8, and include the following theorem. Theorem 1.4. Fix a prime p and suppose x X α (F alg p ). Then Diff( D, α) consists of a single prime p of F, which lies above p. Moreover length(o sh X α,x) = 1 2 ord p(αdp). In particular, the length of the strictly Henselian local ring is independent of x. By this theorem, one sees that when X α is not empty Diff( D, α) = {p} consists of a unique prime, and X α is supported at the rational prime p below p. Moreover, the theorem implies (1.2) deg(x α ) = 1 2 log p ord p(αdp) (E 1,E 2) [X (F alg p )] φ L(E 1,E 2) deg CM (φ)=α 1 Aut(E 1, E 2, φ). We can view the Hecke Eisenstein series as a special case of an incoherent Eisenstein in the sense of Kudla [7]. When Diff( D, α) = {p}, one can also write a α as (see Lemma 3.11) a α = a α (p)b α (p)

4 4 BENJAMIN HOWARD AND TONGHAI YANG where a α (p) is essentially the central derivative of the local Whittaker function at p, and b α (p) is the α-coefficient of a coherent Eisenstein series. Now explicit calculation gives a α (p) = 1 2 log p ord p(αdp). On the other hand, the double sum in (1.2) counts the number of ways to represent α by lattices in the genus of L(E 1, E 2 ) in V (E 1, E 2 ). By the Siegel-Weil formula, it is equal to 1 4 b α(p). This proves Theorem 1.3. In our case, both the double sum and 1 4 b α(p) are easy to compute directly and are equal to ρ(αdp 1 ), thus the explicit formulae in Theorems 1.1 and 1.2. By Theorem 1.3, one sees that the generating function φ(τ) = deg(x α ) q α α D 1 α 0 is the holomorphic part of a (non)-holomorphic Hilbert modular form of weight 1 for SL 2 (O F ), namely E, (τ 1, τ 2, 0). One can also view the theorem as an arithmetic Siegel-Weil formula in the sense of [8] and [9] giving an arithmetic interpretation of the central derivative of the incoherent Eisenstein series. We now explain in what sense Theorems 1.1 and 1.3 are refinements of the earlier work of Gross and Zagier on singular moduli. For a positive integer m let T m be the algebraic stack representing the functor which assigns to a scheme S the category of all triples (E 1, E 2, φ) where (E 1, E 2 ) X (S) and φ Hom(E 1, E 2 ) satisfies deg(φ) = m. Directly from the moduli problems we have T m = X α. α F Tr F/Q (α)=m Combining this decomposition with the formula for deg(x α ) of Theorem 1.1 one finds (see Corollary 2.45) a formula for deg(t m ). This formula is precisely the main result of [3]: Corollary 1.5 (Gross-Zagier). For any positive integer m we have deg(t m ) = 1 2 α D 1 Tr F/Q α=m a 0 log p ord p (αdp)ρ(αdp 1 ) p p p where the middle summation is over those rational primes p which are nonsplit in both K 1 and K 2. Furthermore, deg(t m ) is equal to the m-fourier coefficient of 1 4 E, (τ, τ, 0). This work grew out of the authors attempts to understand Gross and Zagier s work on singular moduli from the perspective of Kudla s program [7, 8, 9] to relate arithmetic intersection multiplicities on Shimura varieties of orthogonal and unitary type to the Fourier coefficients of derivatives of Eisenstein series. On the occasion of his sixtieth birthday the authors wish to express to Steve Kudla both their deepest appreciation for his beautiful mathematics, and their deepest gratitude for his influence on their own lives and work.

5 SINGULAR MODULI REFINED 5 2. Moduli spaces of CM elliptic curves Throughout Section 2 we keep the notation of the introduction, but allow the possibility that gcd(d 1, d 2 ) > 1. For any sets Y X the characteristic function of Y is denoted 1 Y CM pairs. Let S be a scheme and R an order in a quadratic imaginary field. An elliptic curve over S with complex multiplication by R is a pair E = (E, κ) in which E S is an elliptic curve and κ : R End(E) is an action of R on E. Definition 2.1. A CM pair over a scheme S is a pair (E 1, E 2 ) in which E 1 and E 2 are elliptic curves over S with complex multiplication by O K1 and O K2, respectively. An isomorphism between CM pairs (E 1, E 2) (E 1, E 2 ) is a pair (f 1, f 2 ) of isomorphisms of underlying elliptic curves f 1 : E 1 E 1 f 2 : E 2 E 2 which are O K1 and O K2 linear, respectively. To understand the moduli space of CM pairs over schemes we use the language of stacks and groupoids as in [14]. Given a CM pair (E 1, E 2 ) over a scheme S and a morphism of schemes T S there is an evident notion of the pullback CM pair (E 1, E 2 ) /T. Let X be the category whose objects are CM pairs over schemes. In the category X an arrow (E 1, E 2) (E 1, E 2 ) between CM pairs defined over schemes T and S, respectively, is a morphism of schemes T S together with an isomorphism (in the sense of Definition 2.1) of CM pairs over T (E 1, E 2) = (E 1, E 2 ) /T. Thus X is a category fibered in groupoids over the category of schemes. For a scheme S the fiber X (S) is the category of CM pairs over schemes, and arrows in this category are isomorphisms in the sense of Definition 2.1. Definition 2.2. Suppose k is a field of characteristic p > 0. A CM pair (E 1, E 2 ) X (k) is supersingular if the p-divisible groups of the underlying elliptic curves E 1 and E 2 are connected. In other words, the underlying elliptic curves E 1 and E 2 are supersingular in the usual sense. Suppose S is a scheme. For every CM pair (E 1, E 2 ) X (S) we abbreviate L(E 1, E 2 ) = Hom(E 1, E 2 ) (where the Hom means homomorphisms between elliptic curves over S in the usual sense) and V (E 1, E 2 ) = L(E 1, E 2 ) Z Q. Assuming that S is connected the Z-module L(E 1, E 2 ) is equipped with the positive definite quadratic form deg(φ), and we denote by [φ 1, φ 2 ] = deg(φ 1 + φ 2 ) deg(φ 1 ) deg(φ 2 ) = φ 1 φ 2 + φ 2 φ 1 the associated bilinear form. The Q-algebra K = K 1 Q K 2 acts on the Q-vector space V (E 1, E 2 ) by (2.1) (x 1 x 2 ) φ = κ 2 (x 2 ) φ κ 1 (x 1 ).

6 6 BENJAMIN HOWARD AND TONGHAI YANG By a K-Hermitian form on V (E 1, E 2 ) we mean a function, : V (E 1, E 2 ) V (E 1, E 2 ) K which is K-linear in the first variable and satisfies φ 1, φ 2 = φ 2, φ 1. Proposition 2.3. (1) There is a unique F -bilinear form [φ 1, φ 2 ] CM on V (E 1, E 2 ) which satisfies [φ 1, φ 2 ] = Tr F/Q [φ 1, φ 2 ] CM (2) The F -quadratic form deg CM (φ) = 1 2 [φ, φ] CM is the unique F -quadratic form on V (E 1, E 2 ) which satisfies deg(φ) = Tr F/Q deg CM (φ). (3) There is a unique K-Hermitian form φ 1, φ 2 CM on V (E 1, E 2 ) which satisfies [φ 1, φ 2 ] CM = Tr K/F φ 1, φ 2 CM. Proof. Suppose φ 1, φ 2 V (E 1, E 2 ). If x = x 1 x 2 K is nonzero then as elements of End(E 1 ) we have [x φ 1, φ 2 ] = κ 1 (x 1 ) 1 [x φ 1, φ 2 ] κ 1 (x 1 ) = φ 1 κ 2 (x 2 ) φ 2 κ 1 (x 1 ) + κ 1 (x 1 ) φ 2 κ 2 (x 2 ) φ 1 = [φ 1, x φ 2 ]. Thus for all x K we have [x φ 1, φ 2 ] = [φ 1, x φ 2 ]. All of the claims now follow from this property and some elementary linear algebra; in particular from the fact that if M/L is a finite extension of fields then for any finite dimensional M-vector space V the trace Tr M/L induces an isomorphism Hom M (V, M) Hom L (V, L). Thus the complex multiplication structure on E 1 and E 2 endows the set V (E 1, E 2 ) not only with a K-action, but with an F -quadratic form deg CM which refines the usual notion of degree. For every m Q define T m to be the category, fibered in groupoids over schemes, of triples (E 1, E 2, φ) in which (E 1, E 2 ) is a CM pair over a scheme S and φ L(E 1, E 2 ) satisfies deg(φ) = m on every connected component of S. Similarly for every α F define X α to be the category of triples (E 1, E 2, φ) in which (E 1, E 2 ) is a CM pair over a scheme and φ L(E 1, E 2 ) satisfies deg CM (φ) = α on every connected component of S. The categories X, T m, and X α are algebraic (Deligne-Mumford) stacks, in the sense of [14], of finite type over Spec(Z) (briefly, one knows that the category E of elliptic curves over schemes is an algebraic stack of finite type over Spec(Z), and the relative representability of each of X, T m, and X α over E Z E is proved using the methods and results of [5, Chapter 6]). For every m Q there is a decomposition (2.2) T m = X α. α F Tr F/Q (α)=m

7 SINGULAR MODULI REFINED 7 Remark 2.4. For the moment let us write X (d 1, d 2 ) (respectively X α (d 1, d 2 ) ) instead of X (respectively X α ) to emphasize the dependence of X and X α on the quadratic imaginary fields K 1 and K 2. Similarly write F d1,d 2 instead of F. It is clear that the functor (E 1, E 2 ) (E 2, E 1 ) defines an isomorphism of stacks X (d 1, d 2 ) X (d 2, d 1 ). If we identify K 1 Q K 2 = K2 Q K 1 using x y y x then we may identify F d1,d 2 = F = Fd2,d 1, and for any α F the functor (E 1, E 2, φ) (E 2, E 1, φ ) defines an isomorphism of stacks X α (d 1, d 2 ) = X α (d 2, d 1 ). In this sense the stacks X and X α are each symmetric in d 1 and d 2. If S is a scheme and C is any one of X, T m, or X α then [C(S)] denotes the set of isomorphism classes of objects in the category C(S) The support of X α. Given α F define a nondegenerate Q-quadratic form Q α on K by Q α (x) = Tr F/Q (αxx). For each place l of Q let hasse l ( ) be the Hasse invariant on Q l -quadratic spaces and let (, ) l be the usual Hilbert symbol at l. Define the local invariant inv l (α) = ±1 by inv l (α) = hasse l (K l, Q α ) ( 1, 1) l, the modified local invariant of α inv l (α) = and a finite set of places of Q { invl (α) if l < inv l (α) if l =, Sppt(α) = {l inv l (α) = 1}. Note that the product formula l inv l(α) = 1 implies that Sppt(α) has odd cardinality. Lemma 2.5. Suppose that k is an algebraically closed field of nonzero characteristic. If (E 1, E 2 ) X (k) is supersingular then for some totally positive β F there is an isomorphism of F -quadratic spaces (V (E 1, E 2 ), deg CM ) = (K, β Nm K/F ). Proof. Set p = char(k). As all supersingular elliptic curves over k, and all of their endomorphisms, are defined over F alg p, we may assume that k = F alg p. Let H = End(E 1 ) Z Q so that H is a quaternion division algebra over Q of discriminant p. Fix also an isogeny f : E 1 E 2. Then φ f 1 φ f defines an isomorphism of Q-algebras End(E 2 ) Z Q H and φ f φ defines an isomorphism of Q-vector spaces H V (E 1, E 2 ). In particular V (E 1, E 2 ) has dimension 4. Suppose first that d 1 d 2 so that K is a field. Then the dimension of V (E 1, E 2 ) as a K-vector space is 1, and any isomorphism of K-vector spaces V (E 1, E 2 ) = K identifies the Hermitian form, CM on V (E 1, E 2 ) with a Hermitian form on K. All such Hermitian forms have the form x, y = βxy for some β F. As our isomorphism identifies the Q-quadratic

8 8 BENJAMIN HOWARD AND TONGHAI YANG form Q(x) = Tr F/Q (βxx) on K with the positive definite quadratic form deg on V (E 1, E 2 ), it follows easily that β is totally positive. Now suppose that d 1 = d 2 so that F = Q Q and K = K 1 K 2. Fix an isomorphism K 1 = K2, and call this common field K 0. Using the fact that any two embeddings of K 0 into H are conjugate (by the Noether-Skolem theorem), we may adjust the isogeny f in order to assume that f is K 0 -linear. Under the identifications chosen above the embeddings κ i : K 0 End(E i ) Z Q = H are equal and action of the orthogonal idempotents in F = Q Q on V (E 1, E 2 ) = H induces a splitting H = H + H in which H + is the image of K 0 and H = {b H x K 0 xb = bx}. Each of H ± is a one-dimensional K 0 -vector space, and it follows that V (E 1, E 2 ) is free of rank one over K = K 0 Q K 0 = K0 K 0. After fixing an isomorphism of K-modules V (E 1, E 2 ) = K, the remainder of the proof is identical to the case of d 1 d 2. Proposition 2.6. If k is an algebraically closed field of characteristic p 0, α F, and (E 1, E 2, φ) X α (k) then (1) p > 0 and the CM pair (E 1, E 2 ) is supersingular, (2) p is nonsplit in both K 1 and K 2, (3) there are isomorphisms of quadratic spaces over Q (K, Q α ) = (V (E 1, E 2 ), deg) = (H, Nm) where H is the rational quaternion algebra over Q of discriminant p and Nm is the reduced norm on H, (4) Sppt(α) = {p}. Proof. Suppose that p = 0. As φ : E 1 E 2 is a nonzero isogeny, κ 1, κ 2, and φ determine isomorphisms (2.3) K 1 = End(E1 ) Z Q = End(E 2 ) Z Q = K 2 which force d 1 = d 2. Furthermore there are isomorphisms of Q-vector spaces End(E 1 ) Z Q = V (E 1, E 2 ) = End(E 2 ) Z Q, and so V (E 1, E 2 ) is 2-dimensional. As K = K 1 Q K 2 = K1 K 2 one of the two orthogonal idempotents in F = Q Q must annihilate V (E 1, E 2 ). Assuming for simplicity that it is (0, 1) which annihilates V (E 1, E 2 ), the F -quadratic form deg CM is simply (deg, 0). Of course this contradicts deg CM (φ) = α F. Thus k has characteristic p > 0. The existence of the isogeny φ implies that the elliptic curves E 1 and E 2 are either both supersingular of both ordinary. If they are both ordinary then again (2.3) holds and repeating the above argument gives a contradiction. Thus (E 1, E 2 ) is supersingular. In particular End(E 1 ) Z Q = H = End(E 2 ) Z Q. As κ i gives an embedding of K i into H, it follows immediately that p is nonsplit in K i. By Lemma 2.5 there is an isomorphism of F -quadratic spaces (V (E 1, E 2 ), deg CM ) = (K, β Nm K/F )

9 SINGULAR MODULI REFINED 9 for some β F which is determined precisely up to multiplication by an element of Nm K/F (K ). By hypothesis the quadratic space on the left represents α, and so there is u K such that α = β Nm K/F (u). Thus we have an isomorphism of F -quadratic spaces (V (E 1, E 2 ), deg CM ) = (K, α Nm K/F ) and so also an isomorphism of Q-quadratic spaces (V (E 1, E 2 ), deg) = (K, Q α ). Fix an isomorphism of Q-algebras H = End(E 1 ) Z Q. The function f f φ defines an isomorphism of Q-quadratic spaces (V (E 1, E 2 ), deg) = (H, b 1 Nm) where b = deg(φ). As the reduced norm H Q is surjective there is an isomorphism of Q-quadratic spaces (H, b Nm) = (H, Nm). It only remains to prove that Sppt(α) = {p}. Using the isomorphism (K, Q α ) = (H, Nm) already proved we find inv l (α) = hasse l (H l, Nm) ( 1, 1) l. By direct calculation of the Hasse invariant of (H l, Nm) it follows that { 1 if l = p, inv l (α) = 1 otherwise. In particular inv l (α) = 1 if and only if l = p. This completes the proof. Corollary 2.7. Suppose α F and that X α is nonempty. positive. Then α is totally Proof. Suppose that x X α (k) for some algebraically closed field k. By Proposition 2.6 k has characteristic p and x is a supersingular point. By Lemma 2.5 there is a totally positive β F and a u K such that α = β Nm K/F (u). Thus α is totally positive. Corollary 2.8. Suppose α F. If Sppt(α) = {p} for a finite prime p then all geometric points of X α lie in characteristic p and are supersingular. If Sppt(α) > 1 or if Sppt(α) = { } then X α =. Proof. This is immediate from Proposition Group actions. For i = 1, 2 define an algebraic group over Q by T i (A) = (K i Q A) for any Q-algebra A. Let ν i : T i G m be norm ν i (t i ) = t i t i and define T (A) = {(t 1, t 2 ) T 1 (A) T 2 (A) ν 1 (t 1 ) = ν 2 (t 2 )}. Define an algebraic group S over Q by S(A) = {z (K Q A) Nm K/F (z) = 1}. There is an evident character ν : T G m defined by the relations ν 1 (t 1 ) = ν(t) = ν 2 (t 2 ) for t = (t 1, t 2 ) T (R) and a homomorphism T S defined by (2.4) t t 1 t 2 ν(t). Let U T (A f ) be the compact open subgroup U = T (A f ) (Ô K 1 Ô K 2 )

10 10 BENJAMIN HOWARD AND TONGHAI YANG and let V S(A f ) be the image of U under T (A f ) S(A f ). Proposition 2.9. If k is a field of characteristic 0, the ring of adeles A, or the ring of finite adeles A f then the sequence 1 k T (k) S(k) 1 is exact, where k T (k) is the diagonal inclusion and T (k) S(k) is (2.4). Proof. Suppose first that k is a field of characteristic 0. Fix an algebraic closure k alg /k and embeddings of E 1 and E 2 into k alg. There is then an isomorphism of k alg -algebras E i Q k alg = k alg k alg defined by x i 1 (x i, x i ) which we use to identify T i (k alg ) = G 2 m(k alg ). The group T (k alg ) is then identified with T (k alg ) = {(x 1, x 2, y 1, y 2 ) (G 2 m G 2 m)(k alg ) x 1 x 2 = y 1 y 2 }. Recalling that K = K 1 Q K 2 we now identify K Q k alg = k alg k alg k alg k alg using the k alg -isomorphism (x y) 1 (xy, xy, xy, xy). Under this identification S(k alg ) = {(a 1, a 2, b 1, b 2 ) G 4 m(k alg ) a 1 a 2 = 1 = b 1 b 2 } and the map T (k alg ) S(k alg ) takes the form ( x1 (x 1, x 2, y 1, y 2 ), x 2, y 2, y ) 1. y 2 y 1 x 2 x 1 Using these explicit formulae one easily verifies the exactness of 1 G m (k alg ) T (k alg ) S(k alg ) 1, and the claim follows by taking Gal(k alg /k) cohomology and applying Hilbert s Theorem 90. If k = A then the proof is essentially the same: first choose a finite Galois extension M/Q which contains both K 1 and K 2 (e.g. take M = K if d 1 d 2 ) and then, as above, use E i Q A M = AM A M to prove the exactness of 1 A M T (A M ) S(A M ) 1. The adelic form of Hilbert s Theorem 90 [10, Corollary 8.1.3] then proves the exactness of 1 A T (A) S(A) 1, and the exactness for k = A implies the exactness for k = A f. Corollary The homomorphism T S defined by (2.4) induces an isomorphism T (Q)\T (A f )/U = S(Q)\S(A f )/V. Proof. This is immediate from Proposition 2.9 and the observation A f T (Q)U.

11 SINGULAR MODULI REFINED 11 Suppose (E 1, E 2 ) X (k) with k an algebraically closed field. Define an action of S(Q) on V (E 1, E 2 ) by restricting the action (2.1) from K to the subgroup S(Q). The group T (Q) then also acts on V (E 1, E 2 ) via the homomorphism T S defined above, and this action is given by the simple formula (2.5) t φ = κ 2 (t 2 ) φ κ 1 (t 1 ) 1 for t = (t 1, t 2 ) T (Q). Lemma If (E 1, E 2 ) X (F alg p ) is supersingular then the above action of S(Q) on V (E 1, E 2 ) identifies S(Q) with the special orthogonal group of the F -quadratic form deg CM. The same is true if Q is replaced by Q l, F is replaced by F l, and V (E 1, E 2 ) is replaced by V (E 1, E 2 ) Q Q l for any prime l. Proof. By Lemma 2.5 it suffices to show that the special orthogonal group of the F -quadratic space (K, Nm K/F ) is S(Q) = {x K Nm K/F (x) = 1}. This is well-known; for example [6, Corollary V.6.1.3] implies that every orthogonal transformation of (K, Nm K/F ) of determinant 1 is K-linear, so is given by multiplication by an element of S(Q). For i = 1, 2 let Pic(O Ki ) be the ideal class group of K i and set Γ = Pic(O K1 ) Pic(O K2 ). Using the isomorphism Pic(O Ki ) = K i \Ê i /Ô K i and the canonical injection T (Q)\T (A f )/U (K 1 \Ê 1 /Ô K 1 ) (K 2 \Ê 2 /Ô K 2 ) we identify T (Q)\T (A f )/U with a subgroup Γ 0 Γ. To be explicit, Γ 0 is the image of the injection (2.6) T (Q)\T (A f )/U Γ which sends (t 1, t 2 ) T (A f ) to the pair of ideal classes ([a 1 ], [a 2 ]) defined by a i Ô Ki = t i Ô Ki. For any scheme S the group Γ acts on the set [X (S)] on the right by Serre s tensor construction [1, Section 7] (E 1, E 2 ) ([a 1 ], [a 2 ]) = (E 1 OK1 a 1, E 2 OK2 a 2 ). Remark The classical theory of complex multiplication implies that the action of Γ on [X (C)] breaks [X (C)] into a disjoint union of four simply transitive orbits. The orbits are indexed by the set of all ordered pairs (π 1, π 2 ) in which π 1 : K 1 C π 2 : K 2 C are embeddings of fields. The isomorphism class of a CM pair (E 1, E 2 ) X (C) lies in the orbit indexed by (π 1, π 2 ) if and only if the action of K i on the 1-dimensional C-vector space Lie(E i ) is through π i for both i = 1 and i = 2. Lemma Let k be an algebraically closed field. Every x [X (k)] has trivial stabilizer in Γ and satisfies Aut X (k) (x) = O K 1 O K 2. Proof. Suppose we have a pair ([a 1, a 2 ]) Γ and a CM pair (E 1, E 2 ) defined over k with the property that (E 1, E 2 ) = (E 1 OK1 a 1, E 2 OK2 a 2 ).

12 12 BENJAMIN HOWARD AND TONGHAI YANG In particular there is an isomorphism of O Ki -modules and hence, by [1, Lemma 7.14], Hom OKi (E i, E i ) = Hom OKi (E i, E i OKi a i ) End OKi (E i ) = End OKi (E i ) OKi a i. Both as a ring and as an O Ki -module End OKi (E i ) = O Ki, and so a i = OKi as an O Ki -module. Thus a i is a principal ideal. The isomorphism Aut X (k) (x) = O K 1 O K 2 is clear from Aut OKi (E i ) = O K i. Proposition If gcd(d 1, d 2 ) = 1 then Γ 0 = Γ. Proof. For i = 1, 2 fix a fractional O Ki -deal a i, set a i = Nm Ki/Q(a i ), and define a quadratic form on the Q-vector space K i Q i (x) = a i Nm Ki/Q(x). Let W be the Q-vector space K 1 K 2 endowed with the quadratic form Q(x 1, x 2 ) = Q 1 (x 1 ) Q 2 (x 2 ). The claim is that (W, Q) represents 0, and by the Hasse-Minkowski theorem it suffices to prove this everywhere locally. As W Q R has signature (2, 2) it clearly represents 0. Fix a prime l <. The quadratic space W l has discriminant d 1 d 2 Q l /(Q l )2 and Hasse invariant (a 1, d 1 ) l (a 2, d 2 ) l (d 1, d 2 ) l ( 1, 1) l. If d 1 d 2 is not a square in Q l then W l represents 0 by [12, Chapter IV.2.2]. Thus we may assume that d 1 = d 2 up to a square in Q l. As a i is the norm of a fractional ideal in K i,l we may factor a i = u i b i with b i equal to the norm of some element in K l and u i Z p. As we assume that gcd(d 1, d 2 ) = 1, at least one of K 1 and K 2 is unramified at l. Thus u 1 is either a norm from K 1,l or a norm from K 2,l, and so either (u 1, d 1 ) l = 1 or (u 1, d 2 ) l = 1. But (u 1, d 1 ) l = (u 1, d 2 ) l as d 1 = d 2 up to a square. Thus we have (a 1, d 1 ) l = (u 1, d 1 ) l = 1. The same argument shows that (a 2, d 2 ) l = 1, and as (d 1, d 2 ) l = 1 is obvious we find that the Hasse invariant of W l is ( 1, 1) l. Again by [12, Chapter IV.2.2] the quadratic space W l represents 0. Having proved that the quadratic space (W, Q) represents 0, we deduce that there is an m Q which is represented both by Q 1 and by Q 2. Choosing r i K i such that Q 1 (r 1 ) = Q 2 (r 2 ) we see that the fractional ideal b i = a i r i lies in the same ideal class as a i, and that (2.7) Nm K1/Q(b 1 ) = Nm K2/Q(b 2 ). Thus we have proved that every element of Γ has the form ([b 1 ], [b 2 ]) with b 1 and b 2 satisfying (2.7). Now choose t i K i satisfying t i Ô K,i = b iô K,i. The relation (2.7) implies that there is a u Ẑ such that Nm K1/Q(t 1 ) = u Nm K2/Q(t 2 ). The hypothesis gcd(d 1, d 2 ) = 1 implies that Ẑ = Nm /Q(Ô K1 K 1 ) Nm /Q(Ô K2 K 2 ).

13 SINGULAR MODULI REFINED 13 Factoring u as the product of the norm of some v1 1 Ô K 1 and the norm of some v 2 Ô K 2 we may then replace t i by t i v i so that (t 1, t 2 ) T (A f ). This proves the surjectivity of (2.6), and completes the proof that Γ 0 = Γ Orbital integrals. Fix a prime p and a supersingular CM pair (E 1, E 2 ) X (F alg p ). For every prime l < define L l (E 1, E 2 ) = L(E 1, E 2 ) Z Z l V l (E 1, E 2 ) = V (E 1, E 2 ) Q Q l. Given an α F p the orbital integral O l (α, E 1, E 2 ) is defined (using the action (2.5) of T (Q l ) on V l (E 1, E 2 )) to be (2.8) O l (α, E 1, E 2 ) = 1 Ll (E 1,E 2)(t 1 φ) t Q l \T (Q l)/u l if there exists a φ V l (E 1, E 2 ) satisfying deg CM (φ) = α. If no such φ exists then set O l (α, E 1, E 2 ) = 0. Lemma The orbital integral O l (α, E 1, E 2 ) is independent of the choice of φ used in its definition. Furthermore if (E 1, E 2) [X (F alg p )] lies in the same Γ 0 -orbit as (E 1, E 2 ) then (2.9) O l (α, E 1, E 2 ) = O l (α, E 1, E 2). Proof. Combining Lemma 2.11 with the surjectivity (Proposition 2.9) of T (Q l ) S(Q l ) shows that the group T (Q l ) acts transitively on the set of all φ V l (E 1, E 2 ) for which deg CM (φ) = α. The independence of the orbital integral on the choice of φ is an immediate consequence of this. For the second claim, fix a t = (t 1, t 2 ) T (A f ). Set a i = t i Ô Ki and (E 1, E 2) = (E 1 OK1 a 1, E 2 OK2 a 2 ). There is a K i -linear quasi-isogeny f i Hom(E i, E i ) Z Q defined by f i (x) = x 1 The degree of f i is Nm Ki /Q(a i ) 1, and in particular (2.10) deg(f 1 ) = deg(f 2 ). The isomorphism (2.11) V l (E 1, E 2 ) = V l (E 1, E 2) defined by φ f 2 φ f 1 1 identifies L l (E 1, E 2) with the Z l -lattice t L l (E 1, E 2 ) = {κ 2 (t 2 ) φ κ 1 (t 1 ) 1 φ L l (E 1, E 2 )} in V l (E 1, E 2 ) (the action is that of (2.5)). Moreover the isomorphism (2.11) is K l -linear and respects the F l -quadratic form deg CM on source and target (the isomorphism respects the quadratic form deg by (2.10), and the therefore respects deg CM by the uniqueness part of Proposition 2.3) If there is no φ L l (E 1, E 2 ) such that deg CM (φ) = α then the isomorphism (2.11) implies that both sides of (2.9)

14 14 BENJAMIN HOWARD AND TONGHAI YANG are 0. If there is such a φ then O l (α, E 1, E 2) = 1 Ll (E 1,E 2 )(s 1 φ) s Q l \T (Q l)/u l = 1 t Ll (E 1,E 2 )(s 1 φ) s Q l \T (Q l)/u l = O l (α, E 1, E 2 ). Lemma Suppose l is a finite prime different from p which is unramified in at least one of K 1 and K 2, and let δ l F l be any generator of the O F,l -ideal D l. There is an isomorphism of F l -quadratic spaces (V l (E 1, E 2 ), deg CM ) = (K l, δ 1 l Nm Kl /F l ) which is K l -linear and takes the Z l -lattice L l (E 1, E 2 ) isomorphically to O K,l. Proof. The existence of the desired isomorphism for some choice of δ 1 l F l is clear from Lemma 2.5 and the fact that up to isomorphism the only projective rank one module over O K1,l Zl O K2,l = O K,l is O K,l. Such a δ 1 l is uniquely determined up to multiplication by a norm from O K,l. To show that δ lo F,l = D l the essential observation is that the Z l -bilinear form [, ] is a perfect pairing L l (E 1, E 2 ) L l (E 1, E 2 ) Z l. Indeed, any choice of Z l bases for the Tate modules Ta l (E 1 ) and Ta l (E 2 ) determines an isomorphism of Z l -modules L l (E 1, E 2 ) = Hom Zl (Ta l (E 1 ), Ta l (E 2 )) = M 2 (Z l ) which takes the quadratic form deg to the quadratic form u det for some u Z p. After adjusting the choice of basis we may assume that u = 1, and the isomorphism then identifies the bilinear form [φ 1, φ 2 ] on L l (E 1, E 2 ) with the bilinear form [X, Y ] = Tr(XY ι ) on M 2 (Z l ) (where Y Y ι is the involution satisfying Y Y ι = det(y ) for all Y ). This latter bilinear form is a perfect pairing. It follows that the O F,l -bilinear form of Proposition 2.3 [, ] CM : L l (E 1, E 2 ) L l (E 1, E 2 ) D 1 l is also a perfect pairing, and hence the F l -bilinear form δ 1 l Tr Kl /F l (xy) on K l restricts to a perfect pairing O K,l O K,l D 1 l. The hypothesis that l is unramified in either K 1 or K 2 implies that K l /F l is unramified, and so the trace form Tr Kl /F l is a perfect pairing O K,l O K,l O F,l. It follows that δ l O F,l = D l. This gives the desired isomorphism for some choice of δ l which generates D l. As K l /F l is unramified any other generator of D l differs from δ l by a norm from O K,l, and if u = Nm K l /F l (v) for some v O E,l then multiplication by v 1 defines an isomorphism (K l, δ 1 l which preserves the Z l -lattice O K,l. Nm Kl /F l ) = (K l, uδ 1 l Nm Kl /F l )

15 SINGULAR MODULI REFINED 15 Proposition If l is a finite prime not equal to p which is unramified in at least one of K 1 and K 2 then for every α F Proof. Fix an isomorphism O l (α, E 1, E 2 ) = ρ l (αd). (V l (E 1, E 2 ), deg CM ) = (K l, δ 1 l Nm Kl /F l ) as in Lemma Proposition 2.9 implies that (t 1, t 2 ) ν(t) 1 (t 1 t 2 ) defines an isomorphism (2.12) Q l \T (Q l)/u l S(Q l )/V l which allows us to rewrite the orbital integral (2.8) as (2.13) O l (α, E 1, E 2 ) = 1 OK,l (s 1 φ) s S(Q l )/V l where φ K l satisfies Nm Kl /F l (φ) = α δ l. If no such φ exists then O l (α, E 1, E 2 ) = 0. Suppose first that l is inert in both K 1 and K 2, so that O K,l = Zl 2 Z l 2 O F,l = Zl Z l. In this case Q l \T (Q l)/u l = {1} and (2.13) shows that O l (α, E 1, E 2 ) = 1 if there is a φ K l satisfying Nm Kl /F l (φ) = α δ l. Otherwise O l (α, E 1, E 2 ) = 0. It follows that O l (α, E 1, E 2 ) = ρ l (αd) as both sides are equal to 1 if ord w (αδ l ) is even and nonnegative for both places w of F above l, and otherwise both sides are zero. Suppose next that l is inert in K 1 and is ramified in K 2. Then F l /Q l is a ramified field extension and K l /F l is an unramified field extension. Again one has Q l \T (Q l)/u l = {1} and (2.13) shows that O l (α, E 1, E 2 ) = 1 if there is a φ K l satisfying Nm Kl /F l (φ) = α δ l. Otherwise O l (α, E 1, E 2 ) = 0. It follows that O l (α, E 1, E 2 ) = ρ l (αd), as both sides are equal to 1 if ord w (αδ l ) is even and nonnegative for the unique place w of F above l, and otherwise both sides are zero. The case of l ramified in K 1 and inert in K 2 is identical. Suppose next that l is split in K 1 and nonsplit in K 2. Fix an isomorphism O K1,l = Z l Z l and a uniformizer ϖ O K2,l. Let σ be the nontrivial Galois automorphism of K 2,l and define t 1 = (1, Nm K2,l /Q l (ϖ)) K 1,l t 2 = ϖ σ K 2,l. Then Q l \T (Q l)/u l is the infinite cyclic group generated by t = (t 1, t 2 ). identify K l = K1,l Ql K 2,l = K2,l K 2,l via (x 1, x 2 ) y (x 1 y, x 2 y σ ). Then (2.14) F l = {(a, b) K2,l K 2,l a = b} Now and S(Q l ) = {(a, b) K 2,l K 2,l ab = 1}. Using the isomorphism (2.12) and the above generator t Q l \T (Q l)/u l we find that S(Q l )/V l is the infinite cyclic group generated by (ϖ, ϖ 1 ). It now follows from (2.13) that O l (α, E 1, E 2 ) = 1 OK2,l (ϖi φ 1 ) 1 OK2,l (ϖ i φ 2 ) i=

16 16 BENJAMIN HOWARD AND TONGHAI YANG where (φ 1, φ 2 ) K 2,l K 2,l is any element which satisfies (φ 1 φ 2, φ 1 φ 2 ) = αδ l under the identification (2.14). If we let w be the unique place of F above l then O l (α, E 1, E 2 ) = ρ l (αd) as both sides are 1 + ord w (αd) if ord w (αd) 0, and otherwise both sides are zero. The case of l nonsplit in K 1 and split in K 2 is identical. Finally suppose that l is split in both K 1 and K 2 and fix isomorphisms Define K 1 = Ql Q l K 2 = Ql Q l. ρ i,j = (l i, l j ) Q l Q l. The group Q l \T (Q l)/u l is then isomorphic to the quotient of {(ρ a,b, ρ c,d ) K 1 K 2 a + b = c + d} by the subgroup {(ρ a,b, ρ c,d ) K 1 K 2 a = b = c = d}. If we identify (2.15) O K,l = OK1,l O K2,l = Z l Z l Z l Z l via (x 1, x 2 ) (y 1, y 2 ) (x 1 y 1, x 2 y 2, x 1 y 2, x 2 y 1 ) then and O F,l = {(z 1, z 2, z 3, z 4 ) Z l Z l Z l Z l z 1 = z 2, z 3 = z 4 } S(Q l ) = {(z 1, z 2, z 3, z 4 ) Z l Z l Z l Z l z 1 z 2 = 1, z 3 z 4 = 1}. The isomorphism (2.12) takes (ρ a,b, ρ c,d ) to the quadruple (p i, p i, p j, p j ) S(Q l ) where i = c b = a d and j = d b = a c, and a complete set of coset representatives for S(Q l )/V l is given by the set {(p i, p i, p j, p j ) i, j Z}. It now follows from (2.13) that O l (α, E 1, E 2 ) = 1 Zl (ϖ i φ 1 ) 1 Zl (ϖ i φ 2 ) 1 Zl (ϖ j φ 3 ) 1 Zl (ϖ j φ 4 ) <i,j< where (φ 1, φ 2, φ 3, φ 4 ) Z l Z l Z l Z l = OF,l satisfies (φ 1 φ 2, φ 1 φ 2, φ 3 φ 4, φ 3 φ 4 ) = αδ l under (2.15). If we let w 1, w 2 be the two places of F above l then O l (α, E 1, E 2 ) = ρ l (αd) as both sides are (1 + ord w1 (αd))(1 + ord w2 (αd)) if ord w1 (αd) 0 and ord w2 (αd) 0, and otherwise both sides are zero. Proposition For any totally positive α F and any prime p 1 Aut Xα (F alg p ) (x) = 1 O l (α, E 1, E 2 ) 2 x [X α(f alg p )] (E 1,E 2 ) l< where the sum on the right is over the supersingular points (E 1, E 2 ) [X (F alg p )]/Γ 0.

17 SINGULAR MODULI REFINED 17 Proof. Using Lemma 2.13 for the final equality we have 1 Aut Xα (F alg p ) (x) (2.16) x [X α (F alg p )] = = (E 1,E 2 ) [X (F alg p )] φ V (E 1,E 2) supersingular deg CM (φ)=α 1 w 1 w 2 (E 1,E 2 ) ([a 1 ],[a 2 ]) Γ 0 1 L(E1,E 2 )(φ) Aut(E 1, E 2 ) φ V (E 1 a 1,E 2 a 2 ) deg CM (φ)=α 1 L(E1 a 1,E 2 a 2 )(φ) in which the outer sum is over the supersingular points (E 1, E 2 ) [X (F alg p )]/Γ 0. For every pair of ideal classes ([a 1 ], [a 2 ]) Γ 0 fix a t = (t 1, t 2 ) T (A f ) whose image under (2.6) is ([a 1 ], [a 2 ]), and set a i = t i Ô Ki. There are quasi-isogenies f 1 : E 1 E 1 a 1 f 2 : E 2 E 2 a 2 both defined by f i (x) = x 1. The isomorphism V (E 1, E 2 ) = V (E 1 a 1, E 2 a 2 ) defined by φ f 2 φ f 1 1 identifies L(E 1 a 1, E 2 a 2 ) with the Ẑ-lattice t L(E 1, E 2 ) = {κ 2 (t 2 ) φ κ 1 (t 1 ) 1 φ L(E 1, E 2 )} in V (E 1, E 2 ) (the action is that of (2.5)). This gives the first equality in 1 L(E1 a 1,E 2 a 2)(φ) (2.17) ([a 1],[a 2]) Γ 0 φ V (E 1 a 1,E 2 a 2) deg CM (φ)=α = = t T (Q)\T (A f )/U φ V (E 1,E 2 ) deg CM (φ)=α s S(Q)\S(A f )/V φ V (E 1,E 2) deg CM (φ)=α 1 t L(E1,E 2 ) (φ) 1 s L(E1,E 2) (φ). Let us assume that there is some φ 0 V (E 1, E 2 ) for which deg CM (φ 0 ) = α. By Lemma 2.11 the group S(Q) acts simply transitively on the set of all such φ 0. Thus the above sum may be rewritten as 1 (φ) = s L(E1,E 2) 1 s L(E1,E 2) (γ 1 φ 0 ) s S(Q)\S(A f )/V (2.18) φ V (E 1,E 2) deg CM (φ)=α In the final equality we have used s S(Q)\S(A f )/V γ S(Q) = S(Q) V = w 1w 2 2 s S(A f )/V O l (α, E 1, E 2 ). S(Q) V = (T (Q) U)/{±1} = w 1w 2. 2 l 1 s L(E1,E 2 ) (φ 0)

18 18 BENJAMIN HOWARD AND TONGHAI YANG If no such φ 0 exists then both the first and last expression in (2.18) vanish. Combining (2.16), (2.17), and (2.18) completes the proof Local calculations I. Fix a prime p. For a positive integer d let F p d F alg p be the subfield of p d elements and let Z p d = W (F p d) W = W (F alg p ) be the Witt vectors of F p d and F alg p, respectively. Denote by σ : W W the continuous ring automorphism of W which reduces to x x p on the residue field W/pW = F alg p. Let Q p d be the fraction field of Z p d. Hypothesis Throughout Section 2.5 we assume that p is inert in both K 1 and K 2. This hypothesis implies that O K,p = Zp 2 Z p 2 O F,p = Zp Z p but we do not (yet) fix such isomorphisms. This hypothesis also implies that all CM pairs over F alg p are supersingular. Fix a supersingular CM pair (E 1, E 2 ) X (F alg p ). The action of O Ki on Lie(E i ) is through some ring homomorphism π i : O Ki F alg p, and as we assume that p is inert in K i there is a unique ring homomorphism π i : O Ki Z p 2 such that π i is equal to the composition O Ki π i Zp 2 F p 2 F alg p. Recalling that K = K 1 Qp K 2 define a Q-algebra homomorphism ρ : K Q p 2 by ρ(x 1 x 2 ) = π 2 (x 2 ) π 1 (x 1 ). Let q be the prime of E such that ρ factors through the completion ρ : K q Q p 2. Definition The prime p of F lying below q is the reflex prime of the CM pair (E 1, E 2 ) X (F alg p ). Define another Q-algebra homomorphism ρ : K Q p 2 by ρ (x 1 x 2 ) = π 2 (x 2 ) π 1 (x 1 ). Let q be the prime of K above p such that ρ factors though an isomorphism ρ : O K,q Z p 2 and let p be the prime of F below q. One can check that p p. Let g be a connected p-barsotti-tate group of dimension one and height two over F alg p. Up to isomorphism there is a unique such g, and g is isomorphic to the p- Barsotti-Tate group of any supersingular elliptic curve over F alg p. Fix isomorphisms of p-barsotti-tate groups (2.19) f 1 : E 1 [p ] g f 2 : E 2 [p ] g. Set = End(g) so that is the maximal order in a quaternion division algebra over Q p. Fix an embedding of Z p -algebras Z p 2. After possibly pre-composing this embedding with σ : Z p 2 Z p 2 we may assume that the restriction to Z p 2 of the action of on Lie(g) is given by the composition Z p 2 F p 2 F alg p = End (Lie(g)). F alg p

19 SINGULAR MODULI REFINED 19 If we fix a uniformizing parameter Π with the property that xπ = Πx σ for every x Z p 2 then there is a decomposition of Z p -modules (2.20) = Z p 2 Z p 2Π which is orthogonal with respect to the quadratic form Nm (the reduced norm on ). We identify the Z p -modules (2.21) = L p (E 1, E 2 ) via φ f 1 2 φ f 1 and identify = End(E i [p ]) via φ f 1 i φ f i. Lemma The isomorphisms (2.19) may be chosen in such a way that the isomorphism (2.21) identifies the quadratic form deg on L p (E 1, E 2 ) with the quadratic form Nm on, and so that the images of κ i : O Ki,p End(E i [p ]) = for i = 1, 2 are both equal to Z p 2. With these choices the diagram (2.22) O Ki,p is commutative, where the vertical arrow is the inclusion Z p 2. κ i π i Proof. By the Noether-Skolem theorem there is an η i ( Zp Q p ) with the property that η i κ i η 1 i : K i,p Zp Q p has image Q p 2. Multiplying η i on the left by a multiple of Π does not change the image of this embedding, and so we may assume that η i. If we then replace f i by η i f i then the resulting κ i : O Ki,p has image Z p 2. By examining the Dieudonné module of g one can show that g admits a principal polarization g = g which is unique up to multiplication by Z p. After fixing such a polarization the resulting Rosati involution on is equal to the main involution φ φ ι and the automorphism deg(f i ) = f i fi of g lies in Z p (here we identify E i [p ] with its dual p-barsotti-tate group using the canonical principal polarization of E i ). Under the isomorphism (2.21) the quadratic form deg on L p (E 1, E 2 ) is identified with the Z p -quadratic form Z p 2 Q(φ) = (f2 1 φ f 1 ) (f2 1 φ f 1 ) = deg(f 1 ) deg(f 2 ) 1 Nm(φ) on. If we choose a u i Z p such that Nm(u 2 i ) = deg(f i ) 1 and replace f i by u i f i then deg(f i ) = 1, and the isomorphism (2.21) now identifies deg with Nm. As κ i : O Ki,p has image Z p 2, to prove the commutativity of the diagram (2.22) it suffices to prove that the reductions O Ki,p Z p 2 F p 2 of κ i and π i are equal. This follows from our normalization of the inclusion Z p 2, as both reductions agree with the map O Ki,p F alg p = End (Lie(g)) F alg p describing the action of O Ki,p on the Lie algebra of g (identified with the Lie algebra of E i [p ] using the isomorphism f i ).

20 20 BENJAMIN HOWARD AND TONGHAI YANG From now on we assume that f 1 and f 2 have been chosen as in Lemma The isomorphism of Z p -quadratic spaces (2.21) is an isomorphism of O K,p -modules where the action of O K,p = OK1,p Zp O K2,p on is through (2.23) (x 1 x 2 ) φ = κ 2 (x 2 ) φ κ 1 (x 1 ). This action preserves each summand in the decomposition (2.20): the action of x O K,p on the summand Z p 2 is through left multiplication by ρ (x), and the action on the summand Z p 2Π is through left multiplication by ρ(x). Expressed differently, if we identify O K,p = OK,p O K,p then the action of (z, z) O K,p on is through left multiplication by ρ (z ) on the summand Z p 2 and through left multiplication by ρ(z) on the summand Z p 2Π. This shows that is free of rank one over O K,p and has 1 + Π as a basis. Under the identification O K,p = OK,p O K,p = defined by (z, z) ρ (z ) + ρ(z)π the quadratic form Nm on is identified with the quadratic form x Tr Fp /Q p ( β NmKp /F p (x) ) on O K,p where β = (1, Nm(Π)) O F,p O F,p = OF,p. We have proved the existence of a O K,p -linear isomorphism of Z p -quadratic space (2.24) (, Nm) = (O K,p, Tr Fp/Q p β Nm Kp/F p ). Proposition Let β F p be any element satisfying (2.25) ord p (β) = 1 ord p (β) = 0. There is an isomorphism of F p -quadratic spaces (V p (E 1, E 2 ), deg CM ) = (K p, β Nm Kp/F p ) which is K p -linear and takes the Z p -lattice L p (E 1, E 2 ) isomorphically to O K,p. Proof. Combining (2.21) with (2.24) shows that for some β satisfying (2.25) there is an K p -linear isomorphism of Q p -quadratic spaces (V p (E 1, E 2 ), deg) = (K p, Tr Fp /Q p β Nm Kp /F p ) which takes the Z p -lattice L p (E 1, E 2 ) isomorphically to O K,p. Two quadratic forms Q 1 and Q 2 on a finite free F p -module are equal if and only if the Q p -quadratic forms Tr Fp /Q p Q 1 and Tr Fp /Q p Q 2 are equal, and it follows that the above isomorphism is also an isomorphism of F p -quadratic spaces (V p (E 1, E 2 ), deg CM ) = (K p, β Nm Kp/F p ). This proves the claim for some β satisfying (2.25). The claim for all β satisfying (2.25) follows (as in the proof of Lemma 2.16) from the fact that E p /F p is unramified, which implies that the norm map is surjective on units. Proposition For any α F p O p (α, E 1, E 2 ) = ρ p (αdp 1 ).

21 SINGULAR MODULI REFINED 21 Proof. After choosing isomorphisms K 1,p = Qp 2 and K 2,p = Qp 2 one sees that Q p \T (Q p )/U p = {1}, and so by (2.8) the orbital integral O p (α, E 1, E 2 ) is 1 if there is a φ L p (E 1, E 2 ) satisfying deg CM (φ) = α and is 0 otherwise. Using the model of Proposition (2.22) we see that O p (α, E 1, E 2 ) = 1 if and only if there is a φ O K,p satisfying and after choosing isomorphisms Nm Kp /F p (φ) = αβ 1, K p = Qp 2 Q p 2 F p = Qp Q p we see that such a φ exists if and only if ord w (αβ 1 ) is even and nonnegative for both places w of F above p. Using ord w (αβ 1 ) = ord w (αdp 1 ) we find that both sides of the desired equality are 1 if ord w (αdp 1 ) is even and nonnegative for both places w of F above p, and otherwise both sides of the equality are 0. Corollary Suppose that gcd(d 1, d 2 ) = 1. For every α F O l (α, E 1, E 2 ) = ρ(αdp 1 ). l< Proof. This is clear from Proposition 2.17, Proposition 2.23, and (1.1). Return to the action of K p on Zp Q p defined by (2.23). The orthogonal idempotents (1, 0) and (0, 1) in (2.26) F p = Fp F p = Qp Q p act as the projections to the first and second summands, respectively, in and the function Zp Q p = Qp 2 Q p 2Π, Nm CM (a + bπ) = (aa σ, bb σ Nm(Π)) Q p Q p is an F p -quadratic form (using the identification (2.26)) on Zp Q p which satisfies Nm = Tr Fp /Q p Nm CM. Using the isomorphism (2.21) and Lemma 2.21 we see that there is a K p -linear isomorphism of F p -quadratic spaces (2.27) (V p (E 1, E 2 ), deg CM ) = ( Zp Q p, Nm CM ) which takes the Z p -lattice L p (E 1, E 2 ) isomorphically to. Let CLN be the category of complete local Noetherian W -algebras with residue field F alg p. For any Z p -subalgabra O consider the functor Def(g, O) from CLN to the category of sets which assigns to an object A of CLN the set of isomorphism classes deformations of g, with its O-action, to A. More formally, to an object A we assign the set Def(g, O)(A) of isomorphism classes of triples (G, j, ρ) in which G is a p-barsotti-tate group over A, ρ : G /F alg g is an isomorphism from the p special fiber of G to g, and j : O End(G) is an action of O lifting the action of O on g = G. /F alg p

22 22 BENJAMIN HOWARD AND TONGHAI YANG Proposition 2.25 (Gross). The deformation functor Def(g, Z p 2) is represented by W. For any φ = a + bπ with b 0 the deformation functor Def(g, Z p 2[φ]) is represented by W/p m W where (2.28) m = ord p(bb σ ) Proof. By [13, Theorem 3.8] the deformation functor Def(g, Z p 2) is represented by the complete local Noetherian W -algebra W. Let G be the universal deformation of g over W. If we set W k = W/p k W and G k = G /Wk then according to a result of Gross [2, Proposition 3.3] (see also [17, Theorem 1.4]) the reduction map End(G k ) End(g) identifies End(G k ) = Z p 2 + p k 1. An endomorphism φ = a + bπ therefore lifts to an endomorphism of G k if and only if bπ p k 1, or equivalently if and only if ord p (bb σ ) 2k 2. In other words the endomorphism φ lifts to G m but not to G m+1 where m is defined by (2.28). It follows from the rigidity theorem [11, Proposition 2.9] that the functor Def(g, Z p 2[φ]) is represented by a quotient of W, and by the above discussion that quotient is precisely W/p m W. Proposition For any supersingular x = (E 1, E 2 ) X (F alg p ) the completion of the strictly Henselian local ring of X at x is isomorphic to W. Proof. The completion of the strictly Henselian local ring of X at x represents the functor on CLR which assigns to every object A of CLR the set of isomorphism classes deformations of the CM pair (E 1, E 2 ) to A. Using the isomorphisms (2.19) and the Serre-Tate theorem [1, Theorem 3.3] this functor is identified with the deformation functor Def(g, Z p 2) Def(g, Z p 2). This latter functor is represented by W = W W W, by the first part of Proposition Corollary Suppose p is either prime of F above p and let X p [X (F alg p )] denote the set of supersingular points with reflex prime p. Then X p = 2 Γ. Proof. Fix a continuous embedding of Z p -algebras W C p and an isomorphism between the algebraic closures of Q in C p and C. By the theory of complex multiplication every CM pair (E 1, E 2 ) defined over C or C p admits a model over Q alg, and there are canonical bijections [X (C)] = [X (Q alg )] = [X (C p )]. All CM pairs over C p have good reduction modulo p, and so there is a well defined reduction map [X (C p )] [X (F alg p )]. It follows from Proposition 2.26 that the reduction map is a bijection, and that there are canonical bijections [X (C p )] = [X (O Cp )] = [X (W )] = [X (F alg p )]. By Remark 2.12 the set [X (C)] has 4 Γ elements, and so the same is true of [X (F alg p )]. Let p and p be the two primes of F above p. It now suffices to show that the number of supersingular CM pairs (E 1, E 2 ) over F alg p with reflex prime p is equal