MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES

MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES CHIA-FU YU AND JENG-DAW YU Abstract. We show a mass formula for arbitrary supersingular abelian surfaces in characteristic p.. Introduction In [] Chai studied prime-to-p Hecke correspondences on Siegel moduli spaces in characteristic p and proved a deep geometric result about ordinary l-adic Hecke orbits for any prime l p. Recently Chai and Oort gave a complete answer to what this l-adic Hecke orbit can be; see [2]. In this paper we study the arithmetic aspect of supersingular l-adic Hecke orbits in the Siegel moduli spaces, the extreme situation opposite to the ordinary case. In the case of genus g = 2, we give a complete answer to the size of supersingular Hecke orbits. Let p be a rational prime number and g be a positive integer. Let N 3 be a prime-to-p positive integer. Choose a primitive Nth root of unity ζ N Q C and fix an embedding Q Q p. Let A g,,n denote the moduli space over F p of g-dimensional principally polarized abelian varieties with a symplectic level-n structure with respect to ζ N. Let k be an algebraically closed field of characteristic p. For each point x = A 0 = (A 0, λ 0, η 0 in A g,,n (k and a prime number l p, the l-adic Hecke orbit H l (x is defined to be the countable subset of A g,,n (k that consists of points A such that there is an l-quasi-isogeny from A to A 0 that preserves the polarizations (see 2 for definitions. It is proved in Chai [, Proposition ] that the l-adic Hecke orbit H l (x is finite if and only if x is supersingular. Recall that an abelian variety A over k is supersingular if it is isogenous to a product of supersingular elliptic curves; A is superspecial if it is isomorphic to a product of supersingular elliptic curves. A natural question is whether it is possible to calculate the size of a supersingular Hecke orbit. The answer is affirmative, provided that we know its underlying p-divisible group structure explicitly, through the calculation of geometric mass formulas (see Section 2. This is the task of this paper where we examine the p-divisible group structure of some non-superspecial abelian varieties. Let x = (A 0, λ 0 be a g-dimensional supersingular principally polarized abelian varieties over k. Let Λ x denote the set of isomorphism classes of g-dimensional supersingular principally polarized abelian varieties (A, λ over k such that there exists an isomorphism (A, λ[p ] (A 0, λ 0 [p ] of the attached quasi-polarized p- divisible groups; it is a finite set (see [7, Theorem 2. and Proposition 2.2]. Define the mass Mass(Λ x of Λ x as (. Mass(Λ x := Aut(A, λ. (A,λ Λ x Date: August 4, 2009.

2 CHIA-FU YU AND JENG-DAW YU The main result of this paper is computing the geometric mass Mass(Λ x for arbitrary x when g = 2. Let Λ 2,p be the set of isomorphism classes of polarized superspecial abelian surfaces (A, λ with polarization degree deg λ = p 2 over F p such that ker λ α p α p (see 3.. For each member (A, λ in Λ 2,p, the space of degree-p isogenies ϕ : (A, λ (A, λ with ϕ λ = λ over k is a projective line P over k. Write P A to indicate the space of p-isogenies arising from A. This family is studied in Moret-Bailly [6], and also in Katsura-Oort [5]. One defines an F p 2-structure on P using the W (F p 2-structure of M defined by F 2 = p, where M is the covariant Dieudonné module of A and F is the absolute Frobenius. For any supersingular principally polarized abelian surface (A, λ there exist an (A, λ in Λ 2,p and a degree-p isogeny ϕ : (A, λ (A, λ with ϕ λ = λ. The choice of (A, λ and ϕ may not be unique. However, the degree [F p 2(ξ : F p 2] of the point ξ P A (k that corresponds to ϕ is well-defined. In this paper we prove Theorem.. Let x = (A, λ be a supersingular principally polarized abelian surface over k. Suppose that (A, λ is represented by a pair (A, ξ, where A Λ 2,p and ξ P A (k. Then Mass(Λ x = L p 5760, where (p (p 2 + if F p 2(ξ = F p 2, L p = (p 2 (p 4 p 2 if [F p 2(ξ : F p 2] = 2, (p 2 PSL 2 (F p 2 otherwise. Theorem. calculates the cardinality of l-adic Hecke orbits H l (x, as one has (Corollary 2.3 H l (x = Sp 2g (Z/NZ Mass(Λ x. We mention that the function field analogue of Theorem. where supersingular abelian surfaces are replaced by supersingular Drinfeld modules is established in [0]. This paper is organized as follows. In Section 2 we describe the relationship between supersingular l-adic Hecke orbits and mass formulas. We develop the mass formula for the orbits of certain superspecial abelian varieties. In Section 3 we compute the endomorphism ring of any supersingular abelian surface. The proof of the main theorem is given in the last section. 2. Hecke orbits and mass formulas Let g, p, N, l, A g,,n, k be as in the previous section. We work with a slightly bigger moduli space in which the objects are not necessarily equipped with principal polarizations. It is indeed more convenient to work in this setting. Let A g,p,n = m A g,pm,n be the moduli space over F p of g-dimensional abelian varieties together with a p-power degree polarization and a symplectic level-n structure with respect to ζ N. Write A g,p for the moduli stack over F p that parametrizes g-dimensional p-power degree polarized abelian varieties. For any point x = A 0 = (A 0, λ 0, η 0 in A g,p,n (k, the l-adic Hecke orbit H l (x is defined to be the countable subset of A g,p,n (k that consists of points A such that there

MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES 3 is an l-quasi-isogeny from A to A 0 that preserves the polarizations. An l-quasiisogeny from A to A 0 is an element ϕ Hom(A, A 0 Q such that l m ϕ, for some integer m 0, is an isogeny of l-power degree. 2.. Group theoretical interpretation. Assume that x is supersingular. Let G x be the automorphism group scheme over Z associated to A 0 ; for any commutative ring R, the group of its R-valued points is defined by G x (R = {h (End k (A 0 R h h = }, where h h is the Rosati involution induced by λ 0. Let Λ x,n A g,p,n (k be the subset consisting of objects (A, λ, η such that there is an isomorphism ɛ p : (A, λ[p ] (A 0, λ 0 [p ] of quasi-polarized p-divisible groups. Since l-quasiisogenies do not change the associated p-divisible group structure, we have the inclusion H l (x Λ x,n. Proposition 2.. Notations and assumptions as above. ( There is a natural isomorphism Λ x,n G x (Q\G x (A f /K N of pointed sets, where K N is the stabilizer of η 0 in G x (Ẑ. (2 One has H l (x = Λ x,n. Proof. ( This is a special case of [7, Theorem 2. and Proposition 2.2]. We sketch the proof for the reader s convenience. Let A be an element in Λ x,n. As A is supersingular, there is a quasi-isogeny ϕ : A 0 A such that ϕ λ = λ 0. For each prime q (including p and l, choose an isomorphism ɛ q : A 0 [q ] A[q ] of q-divisible groups compatible with polarizations and level structures. There is an element φ q G x (Q q such that ϕφ q = ɛ q for all q. The map A [(φ q ] gives a well-defined map from Λ x,n to G x (Q\G x (A f /K N. It is not hard to show that this is a bijection. (2 The inclusion H l (x Λ x,n under the isomorphism in ( is given by [G x (Q G x (Ẑ(l ]\[G x (Q l G x (Ẑ(l ]/K N G x (Q\G x (A f /K N. Since the group G x is semi-simple and simply-connected, the strong approximation shows that G x (Q G x (A (l f is dense. The equality then follows immediately. Corollary 2.2. Let A i = (A i, λ i, η i, i =, 2, be two supersingular points in A g,p,n (k. Suppose that there is an isomorphism of the associated quasi-polarized p-divisible groups. Then for any prime l pn there is an l-quasi-isogeny ϕ : A A 2 which preserves the polarizations and level structures. Proof. This follows from the strong approximation property for G x that any element φ in the double space G x (Q\G x (A f /K N can be represented by an element in G x (Q l K (l N, where K(l N Gx(Ẑ(l is the prime-to-l component of K N. Recall that we denote by Λ x the set of isomorphism classes of g-dimensional supersingular p-power degree polarized abelian varieties (A, λ over k such that there is an isomorphism (A, λ[p ] (A 0, λ 0 [p ], and define the mass Mass(Λ x of Λ x as Mass(Λ x := Aut(A, λ. (A,λ Λ x

4 CHIA-FU YU AND JENG-DAW YU Similarly, we define Mass(Λ x,n := Aut(A, λ, η. (A,λ,η Λ x,n Corollary 2.3. One has H l (x = Sp 2g (Z/NZ Mass(Λ x. Proof. This follows from H l (x = Λ x,n = Mass(Λ x,n = G x (Z/NZ Mass(Λ x and G x (Z/NZ = Sp 2g (Z/NZ. 2.2. Relative indices. Write G for the automorphism group scheme associated to a principally polarized superspecial point x 0. The group G Q is unique up to isomorphism. This is an inner form of Sp 2g which is twisted at p and (cf. 3. below. For any supersingular point x A g,p (k, we can regard G x (Z p as an open compact subgroup of G (Q p through a choice of a quasi-isogeny of polarized abelian varieties between x 0 and x. Another choice of quasi-isogeny gives rise to a subgroup which differs from the previous one by the conjugation of an element in G (Q p. For any two open compact subgroups U, U 2 of G (Q p, we put µ(u /U 2 := [U : U U 2 ][U 2 : U U 2 ]. Proposition 2.4. Let x, x 2 be two supersingular points in A g,p (k. Then one has Mass(Λ x2 = Mass(Λ x µ(g x (Z p /G x2 (Z p. Proof. See Theorem 2.7 of [7]. 2.3. The superspecial case. Let Λ g denote the set of isomorphism classes of g-dimensional principally polarized superspecial abelian varieties over F p. When g = 2D > 0 is even, we denote by Λ g,p D the set of isomorphism classes of g- dimensional polarized superspecial abelian varieties (A, λ of degree p 2D over F p satisfying ker λ = A[F ], where F : A A (p is the relative Frobenius morphism on A. Write M g := Aut(A, λ, M g := (A,λ Λ g Aut(A, λ (A,λ Λ g,p D for the mass attached to the finite sets Λ g and Λ g,p D, respectively. Theorem 2.5. Notations as above. ( For any positive integer g, one has { g } M g = ( g(g+/2 2 g ζ( 2k k= g { (p k + ( k}, where ζ(s is the Riemann zeta function. (2 For any positive even integer g = 2D, one has { g } D Mg = ( g(g+/2 2 g ζ( 2k (p 4k 2. k= k= k=

MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES 5 Proof. ( This is due to Ekedahl and Hashimoto-Ibukiyama (see [3, p.59] and [4, Proposition 9], also cf. [8, Section 3]. (2 See Theorem 6.6 of [8]. Corollary 2.6. One has M 2 = (p (p2 +, and M2 = (p2 5760 5760. Proof. This follows from Theorem 2.5 and the basic fact ζ( = 2 and ζ( 3 = 20. This is also obtained in Katsura-Oort [5, Theorem 5. and Theorem 5.2] by a method different from above. Remark 2.7. Proposition 2. is generalized to the moduli spaces of PEL-type in [9], with modification due to the failure of the Hasse principle. 3. Endomorphism rings In this section we treat the endomorphism rings of supersingular abelian surfaces. 3.. Basic setting. For any abelian variety A over k, the a-number a(a of A is defined by a(a := dim k Hom(α p, A. Here α p is the kernel of the Frobenius morphism F : G a G a on the additive group. Denote by DM the category of Dieudonné modules over k. If M is the (covariant Dieudonné module of A, then a(a = a(m := dim k M/(F, V M. Let B p, denote the quaternion algebra over Q which is ramified exactly at {p, }. Let D be the division quaternion algebra over Q p and O D be the maximal order. Let W = W (k be the ring of Witt vectors over k, B(k := Frac(W (k the fraction field, and σ the Frobenius map on W (k. We also write Q p 2 and Z p 2 for B(F p 2 and W (F p 2, respectively. Let A be an abelian variety (over any field. The endomorphism ring End(A is an order of the semi-simple algebra End(A Q. Determining End(A is equivalent to determining the semi-simple algebra End(A Q and all local orders End(A Z l. Suppose that A is a supersingular abelian variety over k. We know that End(A Q = M g (B p,, and End(A Z l = M 2g (Z l for all primes l p. Therefore, it is sufficient to determine the local endomorphism ring End(A Z p = End DM (M, which is an order of the simple algebra M g (D. 3.2. The surface case. Let A be a supersingular abelian surface over k. There is a superspecial abelian surface A and an isogeny ϕ : A A of degree p. Let M and M be the covariant Dieudonné modules of A and A, respectively. One regards M as a submodule of M through the injective map ϕ. Let N be the Dieudonné submodule in M Q p such that V N = M. If a(m =, then M = (F, V M and hence it is determined by M. If a(m = 2, or equivalently M is superspecial, then there are p 2 + superspecial submodules M M such that dim k M/M =. Now we fix a rank 4 superspecial Dieudonné module N (and hence fix M and consider the space X of Dieudonné submodules M with M M N and

6 CHIA-FU YU AND JENG-DAW YU dim k N/M =. It is clear that X is isomorphic to the projective line P over k. Let Ñ N be the W (F p 2-submodule defined by F 2 = p. This gives an F p 2-structure on P. It is easy to show the following Lemma 3.. Let ξ P (k be the point corresponding to a Dieudonné module M in X. Then M is superspecial if and only if ξ P (F p 2. Choose a W -basis e, e 2, e 3, e 4 for N such that F e = e 2, F e 2 = pe, F e 3 = e 4, F e 4 = pe 3. Note that this is a W (F p 2-basis for Ñ. Write ξ = [a : b] P (k. The corresponding Dieudonné module M is given by M = Span < pe, pe 3, e 2, e 4, v >, where v = a e + b e 3 and a, b W are any liftings of a, b respectively. Case (i: ξ P (F p 2. In this case M is superspecial. We have End DM (M = M 2 (O D. Assume that ξ P (F p 2. In this case a(m =. If φ End DM (M, then φ End DM (N. Therefore, End DM (M = { φ End DM (N ; φ(m M }. We have End DM (N = End DM (Ñ = M 2(O D. The induced map (3. π : End DM (Ñ EndDM(Ñ/V Ñ is surjective. Put We have Put V 0 := Ñ/V Ñ = F p 2e F p 2e 3 and B 0 := End Fp 2 (V 0. End DM (Ñ/V Ñ = End F p 2 (V 0 = M 2 (F p 2. B 0 := {T B 0 ; T (v k v }, where v = ae + be 3 V 0 Fp 2 k. Therefore, End DM (M = π (B 0. Since ξ P (F p 2, ( a 0. We write ξ = [ : b], v = e +be 3, and we have F p 2(ξ = F p 2(b. a a Write T = 2 B a 2 a 0, where a ij F p 2. From T (v kv, we get the 22 condition (3.2 a 2 b 2 + (a a 22 b a 2 = 0. Case (ii: F p 2(ξ/F p 2 is quadratic. Write ξ = [ : b]. Suppose b satisfies b 2 = αb + β, where α, β F p 2. Plugging this in (3.2, we get a a 2 + a 2 α = 0 and a 2 β = a 2. This shows { ( 0 (3.3 B 0 = t I + t 2 β α where X 2 αx β is the minimal polynomial of b. } ; t, t 2 F p 2 F p 2(ξ,

MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES 7 Case (iii: ξ P (F p 2 and F p 2(ξ/F p 2 is not quadratic. In this case a 2 = a 2 = 0 and a = a 22. We have {( } a 0 B 0 = ; a F 0 a p 2. We conclude Proposition 3.2. Let A be a supersingular surface over k and M be the associated covariant Dieudonné module. Suppose that A is represented by a pair (A, ξ, where A is a superspecial abelian surface and ξ P A (k. Let π : M 2 (O D M 2 (F p 2 be the natural projection. ( If F p 2(ξ = F p 2, then End DM (M = M 2 (O D. (2 If [F p 2(ξ : F p 2] = 2, then End DM (M {φ M 2 (O D ; π(φ B 0 }, where B 0 M 2 (F p 2 is a subalgebra isomorphic to F p 2(ξ. (3 If it is neither in the case ( nor (2, then { ( } a 0 End DM (M φ M 2 (O D ; π(φ =, a F 0 a p 2. 4. Proof of Theorem. 4.. The automorphism groups. Let x = (A, λ be a supersingular principally polarized abelian surfaces over k. Let x = (A, λ be an element in Λ 2,p such that there is a degree-p isogeny ϕ : (A, λ (A, λ of polarized abelian varieties. Write ξ = [a : b] P (k the point corresponding to the isogeny ϕ. We choose an F p 2-structure on P as in 3.2. Let (M,, (M,, be the covariant Dieudonné modules associated to ϕ : (A, λ (A, λ. Let N be the submodule in M Q p such that V N = M, and put, N = p,. One has an isomorphism (N,, N (M,, of quasi-polarized Dieudonné modules. Put U x := G x (Z p = Aut DM (M,,, U x := G x (Z p = Aut DM (M,, = Aut DM (N,, N. Choose a W -basis e, e 2, e 3, e 4 for N such that F e = e 2, F e 2 = pe, F e 3 = e 4, F e 4 = pe 3, e, e 3 N = e 3, e N =, e 2, e 4 N = e 4, e 2 N = p, and e i, e j = 0 for all remaining i, j. The Dieudonné module M is given by M = Span < pe, pe 3, e 2, e 4, v >, where v = a e + b e 3 and a, b W are any liftings of a, b respectively. Case (i: ξ P (F p 2. In this case A is superspecial. One has Λ x = Λ 2 and, by Corollary 2.6, Mass(Λ x = (p (p2 +. 5760 In the remaining of this section, we treat the case ξ P (F p 2. One has U x = {φ U x ; φ(m = M },

8 CHIA-FU YU AND JENG-DAW YU and, by Proposition 2.4 and Corollary 2.6, (4. Mass(Λ x = Mass(Λ x µ(u x /U x = p2 5760 [U x : U x ]. Recall that V 0 = Ñ/V Ñ, which is equipped with the non-degenerate alternating pairing, : V 0 V 0 F p 2 induced from, N. The map (3. induces a group homomorphism π : U x Aut(V 0,, = SL 2 (F p 2. Proposition 4.. The map π above is surjective. The proof is given in Subsection 4.2. Lemma 4.2. One has ker π U x. Proof. Let φ ker π. Write φ(e = e + f, φ(e 3 = e 3 + f 3, where f, f 3 V N. Since M is generated by V N and v, it suffices to check φ(v = v + a f + b f 3 M; this is clear. Case (ii: [F p 2(ξ : F p 2] = 2. By Proposition 3.2 and Lemma 4.2, we have π : U x /U x SL 2 (F p 2/F p 2(ξ, where via the identification (3.3. This shows F p 2(ξ = F p 2(ξ SL 2(F p 2 [U x : U x ] = (p 4 p 2. Case (iii: [F p 2(ξ : F p 2] 3. By Proposition 3.2 and Lemma 4.2, we have π : U x /U x SL 2 (F p 2/{±}. This shows [U x : U x ] = PSL 2 (F p 2. From Cases (i-(iii above and equation (4., Theorem. is proved. 4.2. Proof of Proposition 4.. Write O D = W (F p 2[Π], Π 2 = p, Πa = a σ Π, a W (F p 2. The canonical involution is given by (a + bπ = a σ bπ. With the basis, Π, we have the embedding ( a pb σ O D M 2 (W (F p 2, a + bπ = b a σ. Note that this embedding is compatible with the canonical involutions. With respect to the basis e, e 2, e 3, e 4, an element φ End DM (N can be written as ( aij pb T = (T ij M 2 (O D M 4 (W (F p 2, σ ij T ij = a ij + b ij Π = b ij a σ. ij Since φ preserves the pairing, N, we get the condition in M 4 (Q p 2: ( ( ( (4.2 T t J J T =, J =. J J p Note that ( w 0 Tjiw 0 = Tji, t w 0 = M 2 (Z p 2.

MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES 9 The condition (4.2 becomes ( ( w0 w (4.3 T 0 Since we have ( w 0 w 0 w 0 ( J w 0 ( J = Π ( J Π J ( T = J ( = Π J. M 2 (O D, Lemma 4.3. The group U x is the group of O D -linear automorphisms on ( the standard O D -lattice O D O D which preserve that quaternion hermitian form. 0 Π Π 0 We also write (4.3 as ( (4.4 Π T ΠwT = w, w = M 2 (O D. Notation. For an element T M m (D and n Z, write T (n = Π n T Π n. In particular, if T = (T ij M m (Q p 2 M m (D, then T (n = (Tij σn. If T M m (O D, denote by T M m (F p 2 the reduction of T mod Π. Suppose φ SL 2 (F p 2 is given. Then we must find an element T M 2 (O D satisfying (4.4. We show that there is a sequence of elements T n M 2 (O D for n 0 satisfying the conditions (4.5 (Tn ( wt n w (mod Π n+, T n+ T n (mod Π n+, and T 0 = φ. Suppose there is already an element T n M 2 (O D for some n 0 that satisfies (T n ( wt n w (mod Π n+. Put T n+ := T n + B n Π n+, where B n M 2 (O D, and put X n := (T n ( wt n. Suppose X n w + C n Π n+ (mod Π n+2. One computes that X n+ T ( n wt n + Tn ( wb n Π n+ + (Π n+ Bn ( wt n (mod Π n+2 w + C n Π n+ + T ( n wb n Π n+ + ( n+ Bn (n wt n (n+ Π n+ (mod Π n+2. Therefore, we require an element B n M 2 (O D satisfying C n + T t nwb n + ( n+ B t(n+ n wt (n+ n = 0. Put Y n := T t nwb n. As Y t n = B t nwt n, we need to solve the equation C n + Y n + ( n Y t(n+ n = 0, or equivalently the equation { C n + Y n + Y t( n = 0, if n is even, C n + Y n Y t n = 0, if n is odd. It is easy to compute that Xn = X n (. From this it follows that ( n+ C (n+ n Π n+ C ( n Π n+ (mod Π n+2,

0 CHIA-FU YU AND JENG-DAW YU or simply ( n C t(n n = C ( n. This gives the condition { C t n = C ( n, if n is even, C t n = C n, if n is odd. By the following lemma, we prove the existence of {T n } satisfying (4.5. Therefore, Proposition 4. is proved. Lemma 4.4. Let C be an element in the matrix algebra M m (F p 2. ( If C t = C (, then there is an element Y M m (F p 2 such that C+Y +Y t( = 0. (2 If C t = C, then there is an element Y M m (F p 2 such that C+Y Y t = 0. Proof. The proof is elementary and hence omitted. Remark 4.5. Theorem. also provides another way to look at the supersingular locus S 2 of the Siegel threefold. We used to divide it into two parts: superspecial locus and non-superspecial locus. Consider the mass function M : S 2 Q, x Mass(Λ x. Then the function M divides the supersingular locus S 2 into 3 locally closed subsets that refine the previous one. More generally, we can consider the same function M on the supersingular locus S g of the Siegel modular variety of genus g. The situation definitely becomes much more complicated. However, it is worth knowing whether the following question has the affirmative answer. (Question: Is the map M : S g Q a constructible function? Acknowledgments. The research of C.-F. Yu was partially supported by the grants NSC 97-25-M-00-05-MY3 and AS-98-CDA-M0. The research of J.-D. Yu was partially supported by the grant NSC 97-25-M-002-08. References [] C.-L. Chai, Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli. Invent. Math. 2 (995, 439 479. [2] C.-L. Chai, Hecke orbits on Siegel modular varieties. Geometric methods in algebra and number theory, 7 07, Progr. Math., 235, Birkhauser Boston, 2005. [3] T. Ekedahl, On supersingular curves and supersingular abelian varieties. Math. Scand. 60 (987, 5 78. [4] K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms, J. Fac. Sci. Univ. Tokyo 27 (980, 549 60. [5] T. Katsura and F. Oort, Families of supersingular abelian surfaces, Compositio Math. 62 (987, 07 67. [6] L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur P. Sém. sur les pinceaux de courbes de genre au moins deux (ed. L. Szpiro. Astériques 86 (98, 09 40. [7] C.-F. Yu, On the mass formula of supersingular abelian varieties with real multiplications. J. Australian Math. Soc. 78 (2005, 373 392. [8] C.-F. Yu, The supersingular loci and mass formulas on Siegel modular varieties. Doc. Math. (2006, 449 468. [9] C.-F. Yu, Simple mass formulas on Shimura varieties of PEL-type. To appear in Forum Math. [0] C.-F. Yu and J. Yu, Mass formula for supersingular Drinfeld modules. C. R. Acad. Sci. Paris Sér. I Math. 338 (2004 905 908.

MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES (C.-F. Yu Institute of Mathematics, Academia Sinica, 28 Academia Rd. Sec. 2, Nankang, Taipei, Taiwan, and NCTS (Taipei Office E-mail address: chiafu@math.sinica.edu.tw (J.-D. Yu Department of Mathematics, National Taiwan University, Taipei, Taiwan E-mail address: jdyu@math.ntu.edu.tw