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

Wouter Castryck, Hendrik Hubrechts

Published 2009-02-25, updated 2011-01-31Version 3

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.