Arithmetic Intersection of CM points with the reducible locus on the Siegel moduli space

Kristin Lauter, Microsoft Researchjoint work with

Eyal Goren, McGill University

Class Invariants for Quartic CM fields(2004)

Evil Primes and Superspecial Moduli (2005)

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

Motivation

1) Constructing genus 2 curves over finite fields with a given number of points on its Jacobian/Fq (conjectural bound on denominators of Igusa class polynomials: dK)

2) Generalization of elliptic units to S-units for quartic CM fields, DeShalit-Goren’97 (bound primes in the set S)

Clebsch-Bolza-Igusa invariants

i1 = 2 · 35 Χ10−6Χ12

5

i2 = 2-3 · 33 Ψ4 Χ10−4Χ12

3

i3 = 2-5 · 3 Ψ6 Χ10−3Χ12

2

+ 22 · 3 Ψ4 Χ10−4Χ12

3

X10=const*product of even theta chars

Ψi Eisenstein series, Χ12 mod form wt 12

Divisor of X10

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

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].

Supersingular elliptic curves

E1, E2 supersingular elliptic curves over Fp

alg.

Ri = End(Ei) maximal order in Bp, ∞

ą = Hom(E2,E1), ąV = Hom(E1,E2)

End(E1 × E2) = ( R1 ą )

(ąV R2 )

The embedding problem:

To find a ring embedding: OK End(E1 × E2)

such that the Rosati involution coming from the product polarization induces complex conjugation on OK.

Embedding Theorem:

If the embedding problem can be solved then

p < 16d2(Tr(r))2

Note: write r = α+β√d, then Tr(r)=2α.Let d’ = α2-β2d. Then dK = d2d’.

Our bound is: 64 d2 α2. (worse than dK)

Idea of proof:

1) Write down embedding of √d and √r as matrices with entries in Ri 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 OK

4) ** Elements of small norm (compared to p) in Ri commute!

Abelian varieties with CM by K

K CM field of degree 2g over QS(K) = the set of isomorphism classes

over Qalg of abelian varieties (A, λ, ί)A is an abelian variety of dimension g λ:A AV is a principal polarizationί:OK End(A) is a ring embedding,

And the Rosati involution induces complex conjugation on OK.

Evil primes

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

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 < 16d2(Tr(r))2.

Superspecial points on the Hilbert modular varietyL totally real field of degree g and strict class

number 1.p rational prime, unramified in LP prime of Qalg 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 OL equipped with an OL-linear principal polarization.

Superspecial := isomorphic to a product of supersingular elliptic curves

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 p3 for

all p|p;(3) the discriminant of K over L, dK/L, has norm greater

than N in absolute value;then the reduction map

S(K) SS(L)is surjective.

S(K) > SS(L)

OK = OL[x]/( x2 + bx + c), b, c in OL.

-m = b2 - 4c is a totally negative generator of dK/L.

Bp,L = Bp,∞ L

A in SS(L)R= Centralizer of OL in End(A)

R is a superspecial order in Bp,L (Nicole)

Idea of proof:

Given R=End(A), want an OL-embedding of OK into R.

*Suffices to represent –m by ternary quadratic form on a lattice ΛR

*Cogdell-PS-S: Globally iff locally, if the discriminant is large enough ( (3)).

* Suffices check local conditions ( (2))* Superspecial orders are all loc. conj.

Theorem B.

L totally real field of degree 2 and strict class number 1.

p rational prime. A = A2,1 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 ifdL > M the map

SS(L) SS(A)is surjective.

Idea: SS(L)>SS(A)

L=Q(√d), A in SS(A), A = E2 plus polarization λ

Ibukiyama-Katsura-Oort: description of polarizations

Embed √d into λ-symmetric elements of End(A)= M2(R)

Can do this if dL is large enough.

Corollary.

L totally real field of degree 2 and strict class number 1.

p rational prime, unramified in LSuppose dL > M = M(p) (Theorem B).

Then p is evil for every non-biquadratic quartic CM field K satisfying conditions (1) -(3) of Theorem A.

Comparison with Bruinier-Yang

Our bound: p < 16d2(Tr(r))2

Original conjecture: If p divides the denominator of the norm of Igusa invariants, then p|(dK-x2).

Bruinier-Yang conjecture also implies that bound should be dK