Elliptic curves with a given number of points over finite fields
Document Type
Article
Publication Date
2-1-2013
Abstract
Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average. © 2012 The Author(s).
Publication Title
Compositio Mathematica
Recommended Citation
David, C.,
&
Smith, E.
(2013).
Elliptic curves with a given number of points over finite fields.
Compositio Mathematica,
149(2), 175-203.
http://doi.org/10.1112/S0010437X12000541
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/11511