The distance between superspecial abelian varieties with real multiplication

Evil Primes and Superspecial Moduli

For a quartic primitive CM field K, we say that a rational prime p is evil if at least one of the abelian varieties with CM by K reduces modulo a prime ideal p|p to a product of supersingular elliptic curves with the product polarization. We call such primes evil primes for K. In [GL], we showed that for fixed K, such primes are bounded by a quantity related to the discriminant of the field K. ...

A Gross-Zagier formula for quaternion algebras over totally real fields

We prove a higher dimensional generalization of Gross and Zagier’s theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields K and K′, when the curves are reduced modulo a supersingular prime and its powers. Equi...

Real abelian varieties with complex multiplication

Anderson T-motives Are Analogs of Abelian Varieties with Multiplication by Imaginary Quadratic Fields

An analogy between abelian Anderson T-motives of rank r and dimension n , and abelian varieties over C with multiplication by an imaginary quadratic field K, of dimension r and of signature (n, r − n), permits us to get two elementary results in the theory of abelian varieties. Firstly, we can associate to this abelian variety a (roughly speaking) K-vector space of dimension r in C. Secondly, i...

Complex Multiplication, Rationality and Mirror Symmetry for Abelian Varieties

We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.