The Ramanujan Journal vol:30 issue:2 pages:223-242
Let Fq 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 Fq 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.