On invariants of elliptic curves on average

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 $AB>x(\log{x})^\delta$.