A Cohen-Lenstra phenomenon for elliptic curves

TL;DR

This paper investigates the distribution of primes for which the group of points on elliptic curves modulo p is isomorphic to a given finite abelian group, providing an average-case asymptotic formula under a conjecture.

Contribution

It introduces a new asymptotic formula for counting primes related to elliptic curve groups, assuming a conjecture on prime distribution in short intervals.

Findings

Derived an asymptotic count for primes with specified elliptic curve group structure

Established results on average over families of elliptic curves

Connected the distribution to a Cohen-Lenstra type phenomenon

Abstract

Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula for this problem on average over a family of elliptic curves.