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Elliptic curves with a given number of points over finite fields

Part of: Arithmetic algebraic geometry Multiplicative number theory

Published online by Cambridge University Press:  01 November 2012

Chantal David  and
Ethan Smith
Chantal David
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montréal, Québec H3G 1M8, Canada (email: cdavid@mathstat.concordia.ca)
Ethan Smith
Affiliation:
Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec H3C 3J7, Canada Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI, 49931-1295, USA (email: ethans@mtu.edu)
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Abstract

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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 𝔽p. 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.

Keywords

average orderelliptic curvesprimes in short intervalsBarban–Davenport–Halberstam theoremCohen–Lenstra heuristics

MSC classification

Secondary: 11G05: Elliptic curves over global fields 11N13: Primes in progressions
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 2 , February 2013 , pp. 175 - 203
Copyright
© The Author(s) 2012

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