Average twin prime conjecture for elliptic curves

TL;DR

This paper proves that Koblitz's conjecture, which predicts the distribution of primes related to elliptic curves, holds on average over a family of elliptic curves, using advanced distribution techniques.

Contribution

It establishes the average case validity of Koblitz's conjecture for elliptic curves, a significant step forward in understanding prime distributions in this context.

Findings

Koblitz's conjecture holds on average over a two-parameter family of elliptic curves.

Develops a short average distribution result akin to Barban-Davenport-Halberstam.

Provides evidence supporting the conjecture's validity in a broad average sense.

Abstract

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.