The distribution of the number of points modulo an integer on elliptic curves over finite fields

Abstract

Let \(\mathbb{F}_{q}\) be a finite field, and let b and N be integers. We prove explicit estimates for the probability that the number of rational points on a randomly chosen elliptic curve E over \(\mathbb{F}_{q}\) equals b modulo N. The underlying tool is an equidistribution result on the action of Frobenius on the N-torsion subgroup of E. Our results subsume and extend previous work by Achter and Gekeler.