AVERAGE TWIN PRIME CONJECTURE FOR ELLIPTIC CURVES

被引：24

作者：

Balog, Antal ^{[1
]}

Cojocaru, Alina-Carmen ^{[2
,3
]}

David, Chantal ^{[4
,5
]}

机构：

[1] Hungarian Acad Sci, Alfred Renyi Inst Math, H-1053 Budapest, Hungary

[2] Univ Illinois, Dept Math Stat & Comp Sci, Chicago, IL 60607 USA

[3] Acad Romana, Inst Math Simion Stoilow, Bucharest 010702, Romania

[4] Concordia Univ, Dept Math & Stat, Montreal, PQ H3G 1M8, Canada

[5] Inst Adv Study, Princeton, NJ 08540 USA

来源：

AMERICAN JOURNAL OF MATHEMATICS | 2011年 / 133卷 / 05期

基金：

加拿大自然科学与工程研究理事会;

关键词：

MODULO-P; NUMBER; POINTS; ORDERS;

D O I：

10.1353/ajm.2011.0033

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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 IF 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.

引用

页码：1179 / 1229

页数：51

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