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Link to original content: https://doi.org/10.1007/s11139-012-9444-0
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The distribution of the number of points modulo an integer on elliptic curves over finite fields

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Abstract

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.

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Acknowledgements

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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  1. Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001, Leuven (Heverlee), Belgium

    Wouter Castryck & Hendrik Hubrechts

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  1. Wouter Castryck
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  2. Hendrik Hubrechts
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Correspondence to Wouter Castryck.

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Castryck, W., Hubrechts, H. The distribution of the number of points modulo an integer on elliptic curves over finite fields. Ramanujan J 30, 223–242 (2013). https://doi.org/10.1007/s11139-012-9444-0

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