Local indecomposability of Tate modules of non-CM abelian varieties with real multiplication

Real abelian varieties with complex multiplication

Fields of definition of abelian varieties with real multiplication

Let K be a field, and let K be a separable closure of K. Let C be an elliptic curve over K. For each g in the Galois group G := Gal(K/K), let C be the elliptic curve obtained by conjugating C by g. One says that C is an elliptic K-curve if all the elliptic curves C are K-isogenous to C. Recall that a subfield L of K is said to be a (2, . . . , 2)-extension of K if L is a compositum of a finite ...

Moduli of CM abelian varieties

We discuss CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CM-lifted to characteristic zero? Does there exist an abelian variety of dimension g > 3 not isogenous with the Jacobian of an algebraic curve? Can we construct algebraic curves, say over C, where the Jacobian is a

The distance between superspecial abelian varieties with real multiplication

Article history: Received 17 June 2008 Communicated by B. Conrad MSC: primary 14K02 secondary 11E10

Generic Abelian Varieties with Real Multiplication Are Not Jacobians