Secondary terms in the first moment of $\mathrm{\mid }{\mathrm{S}\mathrm{e}\mathrm{l}}_2(E)\mathrm{\mid }$

We prove the existence of secondary terms of order $X^{3\mathrm{/}4}$, with power saving error terms, in the counting functions of $\mathrm{\mid }{\mathrm{S}\mathrm{e}\mathrm{l}}_2(E)\mathrm{\mid }$, 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 $o(X^{5\mathrm{/}6})$, proved by Bhargava–Shankar, where the primary term of order $X^{5\mathrm{/}6}$ for this counting function was obtained.