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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.

Average twin prime conjecture for elliptic curves