Average twin prime conjecture for elliptic curves

By Antal Balog, Alina Carmen Cojocaru, Chantal David. 2007

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Let $E$ be an elliptic curve over ${\Bbb Q}$. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes $p$ up to $x$ such that the order of the group of points of $E$ over ${\Bbb F}_p$ is prime. This is an analogue of the Hardy--Littlewood twin prime conjecture in the case of elliptic curves Koblitz's conjecture is still widely open. In this paper we prove that Koblitz's conjecture is true on average over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result of primes in the style of Barban-Davenport-Halberstam, where the average is taken over prime differences and over arithmetic progressions.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f7b2e92386750292f634cafe99b7827b0a08be1

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In Algebra and Number Theory

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Citations in the corpus

Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height (2006, methodology): “emark that the distribution of prime values of #E a,b(F p) is of great interest in the theory of cryptography. Our method can be adapted to study this question as aand bvary over short intervals; see [6], where also the challenging task of evaluating the main term has been treated. For the problems ”