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)
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 Sunits for quartic CM fields, DeShalitGoren’97 (bound primes in the set S)
Clebsch-Bolza-Igusa invariants
i1 = 2 · 35 Χ10−6Χ125
i2 = 2-3 · 33 Ψ4 Χ10−4Χ123
i3 = 2-5 · 3 Ψ6 Χ10−3Χ122
+ 22 · 3 Ψ4 Χ10−4Χ123
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
Fpalg.
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 Q
S(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 polarization
ί: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 variety
L totally real field of degree g and strict class
number 1.
p rational prime, unramified in L
P 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 if
dL > 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 L
Suppose 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