Secondary terms in the first moment of $|{\rm Sel}_2(E)|$

TL;DR

This paper establishes the existence of secondary terms of order $X^{3/4}$ with power-saving errors in counting 2-Selmer groups of elliptic curves, improving previous error bounds.

Contribution

It introduces the first secondary term of order $X^{3/4}$ in the asymptotic count of 2-Selmer groups, refining earlier error estimates.

Findings

Secondary terms of order $X^{3/4}$ are proven to exist.

Power saving error terms are achieved.

Improves upon previous error bounds of $o(X^{5/6})$.

Abstract

We prove the existence of secondary terms of order $X^{3/4}$, with power saving error terms, in the counting functions of $|{\rm Sel}_2(E)|$, the 2-Selmer group of E, for elliptic curves E having height bounded by X. This is the first improvement on the error term of $o(X^{5/6})$, proved by Bhargava--Shankar, where the primary term of order $X^{5/6}$ for this counting function was obtained.