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

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 $m$.

References in corpus (3)

• The Lang-Trotter Conjecture on Average
• Average twin prime conjecture for elliptic curves
• A Remark on the Conjectures of Lang-Trotter and Sato-Tate on Average

Cited by in corpus (4)

• A Remark on the Conjectures of Lang-Trotter and Sato-Tate on Average
• On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields
• Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average
• The Sato-Tate Conjecture on Average for Small Angles