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

Abstract

We prove the existence of secondary terms of order $X^{3/4}$, with power saving error terms, in the counting functions of $|{\mathrm{Sel}}_2(E)|$, 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.