The Distribution of J-invariants for CM Elliptic Curves defined over Zp

Andrew Fiori
Thu, Jan 28, 2016
PIMS, University of Calgary
PIMS CRG in Explicit Methods for Abelian Varieties

Let K be a quadratic imaginary field, and p be a prime which is inert in K. It is known that the mod p reductions of the j-invariants of elliptic curves defined over the algebraic closure of Qp which admit CM by an order of K are equidistributed among the supersingular values in F{p2}. By contrast, if we replace this algebraically closed field by Qp, the j-invariants for many natural families of orders do not share this same distribution and are simply not uniformly distributed among all the supersingular values in Fp.

In this talk I will explain why this occurs, and some of the computations which led me to consider this question.