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Elliptic curves with a given number of points over finite fields

Part of: Multiplicative number theory Arithmetic algebraic geometry

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

References

[BCD11]Balog, A., Cojocaru, A.-C. and David, C., Average twin prime conjecture for elliptic curves, Amer. J. Math. 133 (2011), 1179–1229.CrossRefGoogle Scholar
[BBIJ05]Battista, J., Bayless, J., Ivanov, D. and James, K., Average Frobenius distributions for elliptic curves with nontrivial rational torsion, Acta Arith. 119 (2005), 81–91.CrossRefGoogle Scholar
[Bur63]Burgess, D. A., On character sums and L-series. II, Proc. Lond. Math. Soc. (3) 13 (1963), 524–536.CrossRefGoogle Scholar
[CFJKP11]Calkin, N., Faulkner, B., James, K., King, M. and Penniston, D., Average Frobenius distributions for elliptic curves over abelian extensions, Acta Arith. 149 (2011), 215–244.CrossRefGoogle Scholar
[CL84a]Cohen, H. and Lenstra, H. W. Jr., Heuristics on class groups, in Number theory (New York, 1982), Lecture Notes in Mathematics, vol. 1052 (Springer, Berlin, 1984), 26–36.Google Scholar
[CL84b]Cohen, H. and Lenstra, H. W. Jr., Heuristics on class groups of number fields, in Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Mathematics, vol. 1068 (Springer, Berlin, 1984), 33–62.CrossRefGoogle Scholar
[CIJ]Cojocaru, A. C., Iwaniec, H. and Jones, N., The Lang–Trotter conjecture on Frobenius fields, Preprint.Google Scholar
[Dav80]Davenport, H., Multiplicative number theory, Graduate Texts in Mathematics, vol. 74, second edition (Springer, New York, 1980).CrossRefGoogle Scholar
[DP99]David, C. and Pappalardi, F., Average Frobenius distributions of elliptic curves, Int. Math. Res. Not. IMRN 1999 (1999), 165–183.CrossRefGoogle Scholar
[DP04]David, C. and Pappalardi, F., Average Frobenius distribution for inerts in , J. Ramanujan Math. Soc. 19 (2004), 181–201.Google Scholar
[DS12]David, C. and Smith, E., A Cohen–Lenstra phenomenon for elliptic curves, Preprint (2012), arXiv:1206.1585.Google Scholar
[FR96]Fouvry, E. and Ram Murty, M., On the distribution of supersingular primes, Canad. J. Math. 48 (1996), 81–104.CrossRefGoogle Scholar
[GS03]Granville, A. and Soundararajan, K., The distribution of values of L(1,χ d), Geom. Funct. Anal. 13 (2003), 992–1028.CrossRefGoogle Scholar
[HW79]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, fifth edition (The Clarendon Press, Oxford University Press, New York, NY, 1979).Google Scholar
[IK04]Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[Jam04]James, K., Average Frobenius distributions for elliptic curves with 3-torsion, J. Number Theory 109 (2004), 278–298.CrossRefGoogle Scholar
[JS11]James, K. and Smith, E., Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals, Math. Proc. Cambridge Philos. Soc. 150 (2011), 439–458.CrossRefGoogle Scholar
[Kob88]Koblitz, N., Primality of the number of points on an elliptic curve over a finite field, Pacific J. Math. 131 (1988), 157–165.CrossRefGoogle Scholar
[Kow06]Kowalski, E., Analytic problems for elliptic curves, J. Ramanujan Math. Soc. 21 (2006), 19–114.Google Scholar
[LT76]Lang, S. and Trotter, H., Distribution of Frobenius automorphisms in GL2-extensions of the rational numbers, in Frobenius distributions in GL2-extensions, Lecture Notes in Mathematics, vol. 504 (Springer, Berlin, 1976).CrossRefGoogle Scholar
[LPZ10]Languasco, A., Perelli, A. and Zaccagnini, A., On the Montgomery–Hooley theorem in short intervals, Mathematika 56 (2010), 231–243.CrossRefGoogle Scholar
[Len87]Lenstra, H. W. Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), 649–673.Google Scholar
[MV07]Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97 (Cambridge University Press, Cambridge, 2007).Google Scholar