Frobenius Fields and Frobenius Rings of Elliptic Curves
by
Chantal David
Concordia University
Coauthors: Jorge Jimenez Urroz (UPC, Barcelona) and Nathan Jones (Montreal)

Let E be an elliptic curve over Q. For each prime p of good reduction, E reduces to a curve over F_p, the Frobenius endormorphism of E/F_p satisfies x^2 - a_p(E) x + p, and the Frobenius ring Z[sqrt(a_p^2-4p)] is a subring of the endomorphism ring End(E/F_p). The distribution of the endomorphism rings End(E/F_p) and the endormorphism fields Q(sqrt(a_p^2-4p)) when p varies is a difficult problem. For example, there are no known examples of an elliptic curve over Q where a given quadratic imaginary field K happen infinitely often as the field of endomorphisms Q(sqrt(a_p^2-4p)). We will give in this talk some evidence for the conjectural distributions for the fields and rings of endomorphisms, by showing that we can prove those conjectural distributions averaging over all elliptic curves in a box.

Date received: April 14, 2008