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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{q}$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{q}$\end{document} 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.