Sums of Euler products and statistics of elliptic curves

TL;DR

This paper develops a general framework using Euler products to analyze statistical properties of elliptic curves over finite fields, reprove known conjectures, and explore new questions like amicable pairs.

Contribution

It introduces a flexible method to compute averages of Euler products, enabling reproof of classical conjectures and investigation of new elliptic curve statistics.

Findings

Reproved average Lang-Trotter, Koblitz, and Sato-Tate conjectures.

Computed statistics for amicable pairs and aliquot cycles.

Established a general theorem connecting averages of Euler products to their factors.

Abstract

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average Lang-Trotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.