On invariants of elliptic curves on average

TL;DR

This paper establishes average-case results for invariants of elliptic curves over finite fields, demonstrating asymptotic formulas for sums involving the group exponents of reductions modulo primes, under certain size conditions.

Contribution

It provides new asymptotic formulas for average invariants of elliptic curves over finite fields, extending previous results to larger families with minimal size restrictions.

Findings

Asymptotic formulas for average exponents of elliptic curve reductions

Conditions on family size for the results to hold

Method improves previous bounds on family sizes

Abstract

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides $A$ and $B$. As an example, let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p$ be a prime of good reduction for $E$. Let $e_{E}(p)$ be the exponent of the group of rational points of the reduction modulo $p$ of $E$ over the finite field $\mathbb{F}_p$. Let $\mathcal{C}$ be the family of elliptic curves $$E_{a,b}:~y^2=x^3+ax+b,$$ where $|a|\leq A$ and $|b|\leq B$. We prove that, for any $c>1$ and $k\in \mathbb{N}$, $$\frac{1}{|\mathcal{C}|} \sum_{E\in \mathcal{C}} \sum_{p\leq x} e_E^k(p) = C_k {\rm li}(x^{k+1})+O\left(\frac{x^{k+1}}{(\log{x})^c} \right),$$ as $x\rightarrow \infty$, as long as $A, B>\exp\left(c_{1} (\log{x})^{1/2} \right)$ and $AB>x(\log{x})^{4+2c}$, where $c_1$ is a suitable positive constant. Here $C_k$ is an explicit constant given in the paper which depends only on $k$, and ${\rm li}(x)=\int_{2}^x dt/\log{t}$. We prove several similar results as corollaries to a general theorem. The method of the proof is capable of improving some of the known results with $A, B>x^\epsilon$ and $AB>x(\log{x})^\delta$ to $A, B>\exp\left(c_1 (\log{x})^{1/2} \right)$ and $AB>x(\log{x})^\delta$.