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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem

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Abstract.

Let E be an elliptic curve defined over and of conductor N. For a prime we denote by the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = p E for which the group is cyclic and we show that, under the generalized Riemann hypothesis, p E = ((log N)4 + ɛ) if E is without complex multiplication, and p E = ((log N)2 + ɛ) if E is with complex multiplication, for any 0 < ɛ < 1.

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Authors and Affiliations

  1. Pinceton University, Mathematics Department, 810 Fine Hall, Washington Road, Princeton, NJ, 08540, USA

    A. C. Cojocaru

  2. Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

    M. R. Murty

Authors
  1. A. C. Cojocaru
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  2. M. R. Murty
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Correspondence to M. R. Murty.

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Mathematics Subject Classification (2001):11G05, 11N36, 11R45

Research supported in part by an Ontario Graduate Scholarship.

Research supported in part by an NSERC grant.

Revised version: 11 April 2004

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Cojocaru, A., Murty, M. Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem. Math. Ann. 330, 601–625 (2004). https://doi.org/10.1007/s00208-004-0562-x

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  • DOI: https://doi.org/10.1007/s00208-004-0562-x

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