Average twin prime conjecture for elliptic curves

Balog, Antal; Cojocaru, Alina; David, Chantal

Average twin prime conjecture for elliptic curves

Authors

Balog, Antal

Cojocaru, Alina

David, Chantal

Description

Let E be an elliptic curve over Q. In 1988, 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 the finite field F_p is prime. This is an analogue of the Hardy and 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 in the style of Barban-Davenport-Halberstam, where the average is taken over twin primes and their differences.
28 pages. See also http://www.mathstat.concordia.ca/faculty/cdavid