Point counting on reductions of CM elliptic curves

Point Counting on Reductions of Cm Elliptic Curves

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM Q-curves in certain cases. This generalizes earlier results of Gross, Stark, and others.

Point Counting on Elliptic and Hyperelliptic Curves

Group Order Formulas for Reductions of CM Elliptic Curves

We give an overview of joint work with Karl Rubin on computing the number of points on reductions of elliptic curves with complex multiplication, including some of the history of the problem.

Reductions of Points on Elliptic Curves

Let E be an elliptic curve defined over Q. Let Γ be a subgroup of rank r of the group of rational points E(Q) of E. For any prime p of good reduction, let Γ̄ be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have |Γ̄| ≥ p f(p) , where f(x) is any function such that f(x) → ∞, at an arbitrary slow ...

Products of CM elliptic curves