The Ramanujan Journal

, Volume 30, Issue 2, pp 223–242

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


DOI: 10.1007/s11139-012-9444-0

Cite this article as:
Castryck, W. & Hubrechts, H. Ramanujan J (2013) 30: 223. doi:10.1007/s11139-012-9444-0


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.


Elliptic curvesFinite fieldsFrobenius statisticsModular curves

Mathematics Subject Classification


Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Departement WiskundeKatholieke Universiteit LeuvenLeuven (Heverlee)Belgium