The Ramanujan Journal

, Volume 30, Issue 2, pp 223–242 | Cite as

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



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 curves Finite fields Frobenius statistics Modular curves 

Mathematics Subject Classification

14H52 14K10 



The authors are very grateful to the anonymous referee of a prior submission of this document, to the anonymous referee of the current submission, to Hendrik W. Lenstra for suggesting the use of Chebotarev’s density theorem, and to Barry Mazur and Bjorn Poonen for their helpful comments on modular curves. Both authors thank F.W.O.-Vlaanderen for its financial support. The first author thanks the Massachusetts Institute of Technology for its hospitality.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

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

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