Average twin prime conjecture for elliptic curves

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 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.

Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

“…Our method can be adapted to study this question as a and b vary over short intervals; see [6], where also the challenging task of evaluating the main term has been treated.…”

Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects

“…Some of these results also rely on the GRH and other number theoretic conjectures. However, on average over a and b for the family of curves E a,b with coefficients in |a| ≤ A, |b| ≤ B quite strong unconditional results about the frequency of prime cardinalities have been obtained by Balog, Cojocaru and David [15].…”

“…Square-free values of t 2 p −4p have been studied by David and Jiménez Urroz [49] (using some results of [15,18]) on average over primes p and also over the family of curves E a,b with coefficients in |a| ≤ A, |b| ≤ B.…”

“…Shparlinski and Sutantyo [139] have used a different approach to count, also under the Bateman and Horn Conjecture, isomorphism classes corresponding to elliptic twins. Furthermore, the same approach has also been used in [139] to give an alternative derivation of (15) that is based on a slightly different assumption (thus it can be considered as one more supporting evidence of the validity of (15)). …”

Geometry and arithmetic of verbal dynamical systems on simple groups

“…To find such .g 1 ; : : : ; g s /, we regard them as s extra variables and obtain a self-map as in (1).…”