Analytic problems for elliptic curves

TL;DR

This paper explores analytic number theory problems related to elliptic curves, focusing on their group structures over finite fields and drawing analogies with classical prime distribution questions, revealing interesting distribution patterns.

Contribution

It introduces new local results on the distribution of elliptic curve group structures over finite fields, highlighting a dichotomy in their occurrence.

Findings

Distribution of elliptic curve groups exhibits a dichotomy.

Analogies with prime distribution in arithmetic progressions.

New local results on elliptic curve structures.

Abstract

We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and of the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (Note : This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv).