Average Twin Prime Conjecture for Elliptic Curves
نویسنده
چکیده
Let E be an elliptic curve over 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 Fp 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 the prime differences and over the arithmetic progressions.
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