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

TL;DR

This paper derives average asymptotic formulas for prime-related properties of elliptic curves with small coefficients, focusing on Sato--Tate distribution, cyclicity, and divisibility of points.

Contribution

It provides new average asymptotic results for prime properties of elliptic curves with small height, extending understanding of their statistical behavior.

Findings

Asymptotic formulas for prime counts with specific properties

Average behavior of elliptic curves related to Sato--Tate conjecture

Results on cyclicity and divisibility statistics

Abstract

We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with $|a|\le A$ and $|b| \le B$, where $A$ and $B$ are small relative to $x$. Specifically, we investigate behavior with respect to the Sato--Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer $m$.