1. Arithmetic Intersection of CM points with the reducible locus on the Siegel moduli space Kristin Lauter, Microsoft Research joint work with Eyal Goren, McGill University Class Invariants for Quartic CM fields(2004) Evil Primes and Superspecial Moduli (2005)

2. Question: A polarized abelian surface with CM by K For which p is A reducible? i.e. A ≈ E x E’ mod p with product polarization

3. Motivation 1)Constructing genus 2 curves over finite fields with a given number of points on its Jacobian/F q (conjectural bound on denominators of Igusa class polynomials: d K ) 2)Generalization of elliptic units to S- units for quartic CM fields, DeShalit- Goren’97 (bound primes in the set S)

4. Clebsch-Bolza-Igusa invariants i 1 = 2 · 3 5 Χ 10 −6 Χ 12 5 i 2 = 2 -3 · 3 3 Ψ 4 Χ 10 −4 Χ 12 3 i 3 = 2 -5 · 3 Ψ 6 Χ 10 −3 Χ 12 2 + 2 2 · 3 Ψ 4 Χ 10 −4 Χ 12 3 X 10 =const*product of even theta chars Ψ i Eisenstein series, Χ 12 mod form wt 12

5. Divisor of X 10 Locus of reducible polarized abelian surfaces (isomorphic to a product of elliptic curves with the product polarization) Primes appearing in the factorization of the (norms of) denominators are primes where A ≈ E x E’ mod p with the product polarization

6. CM field  K be a primitive quartic CM field  L totally real subfield.  Write L = Q(√d), d > 0 square free.  Write K = L(√r) with r a totally negative element in Z[√d].

7. Supersingular elliptic curves E 1, E 2 supersingular elliptic curves over F p alg. R i = End(E i ) maximal order in B p, ∞ ą = Hom(E 2,E 1 ), ą V = Hom(E 1,E 2 ) End(E 1 × E 2 ) = ( R 1 ą ) (ą V R 2 )

8. The embedding problem: To find a ring embedding: O K  End(E 1 × E 2 ) such that the Rosati involution coming from the product polarization induces complex conjugation on O K.

9. Embedding Theorem: If the embedding problem can be solved then p < 16d 2 (Tr(r)) 2 Note: write r = α+β√d, then Tr(r)=2α. Let d’ = α 2 -β 2 d. Then d K = d 2 d’. Our bound is: 64 d 2 α 2. (worse than d K )

10. Idea of proof: 1)Write down embedding of √d and √r as matrices with entries in R i and ą 2)Entries have norm bounded in terms of discriminant of K 3)Rosati involution induced by the product polarization acts like complex conjugation on O K 4)** Elements of small norm (compared to p) in R i commute!

11. Abelian varieties with CM by K K CM field of degree 2g over Q S(K) = the set of isomorphism classes over Q alg of abelian varieties (A, λ, ί) A is an abelian variety of dimension g λ:A  A V is a principal polarization ί:O K  End(A) is a ring embedding, And the Rosati involution induces complex conjugation on O K.

12. Evil primes A rational prime is evil for K if for some prime P of Q alg there is an element of S(K) whose reduction modulo P is the product of two supersingular elliptic curves with the product polarization.

13. Bound on evil primes Corollary: K a non-biquadratic quartic CM field written as K = Q(√d)(√r). If p evil for K, then p < 16d 2 (Tr(r)) 2.

14. Superspecial points on the Hilbert modular variety L totally real field of degree g and strict class number 1. p rational prime, unramified in L P prime of Q alg above p. SS(L) = superspecial points on the reduction modulo P of the Hilbert modular variety associated to L that parameterizes abelian varieties with real multiplication by O L equipped with an O L -linear principal polarization. Superspecial := isomorphic to a product of supersingular elliptic curves

15. Theorem A. There exists a constant N =N(p,L) such that for every CM field K satisfying: (1)K + = L; (2)Let p be a prime of L above p and P a prime of K above p. (a) If p ≠ 2 then f(P/p) + f(p/p) is odd for all P|p|p; (b) If p = 2 then 3m is a quadratic residue modulo p 3 for all p|p; (3) the discriminant of K over L, d K/L, has norm greater than N in absolute value; then the reduction map S(K)  SS(L) is surjective.

16. S(K)  > SS(L) O K = O L [x]/( x 2 + bx + c), b, c in O L. -m = b 2 - 4c is a totally negative generator of d K/L. B p,L = B p,∞  L A in SS(L) R= Centralizer of O L in End(A) R is a superspecial order in B p,L (Nicole)

17. Idea of proof: Given R=End(A), want an O L -embedding of O K into R. *Suffices to represent –m by ternary quadratic form on a lattice Λ R *Cogdell-PS-S: Globally iff locally, if the discriminant is large enough (  (3)). * Suffices check local conditions (  (2)) * Superspecial orders are all loc. conj.

18. Theorem B. L totally real field of degree 2 and strict class number 1. p rational prime. A = A 2,1 the moduli space of principally polarized abelian surfaces. SS(A) = superspecial points of A (mod p). There exists a constant M = M(p) such that if d L > M the map SS(L)  SS(A) is surjective.

19. Idea: SS(L)  >SS(A) L=Q(√d), A in SS(A), A = E 2 plus polarization λ Ibukiyama-Katsura-Oort: description of polarizations Embed √d into λ-symmetric elements of End(A)= M 2 (R) Can do this if d L is large enough.

20. Corollary. L totally real field of degree 2 and strict class number 1. p rational prime, unramified in L Suppose d L > M = M(p) (Theorem B). Then p is evil for every non-biquadratic quartic CM field K satisfying conditions (1) -(3) of Theorem A.

21. Comparison with Bruinier-Yang Our bound: p < 16d 2 (Tr(r)) 2 Original conjecture: If p divides the denominator of the norm of Igusa invariants, then p|(d K -x 2 ). Bruinier-Yang conjecture also implies that bound should be d K