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

TL;DR

This paper provides explicit probability estimates for the number of rational points on elliptic curves over finite fields modulo an integer, using equidistribution results related to Frobenius actions.

Contribution

It introduces new explicit estimates for point distributions on elliptic curves modulo N, extending previous research by Achter and Gekeler.

Findings

Explicit probability bounds for point counts modulo N

Extension of previous equidistribution results

General framework applicable to various finite fields

Abstract

Let F 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 F 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.