The geometry of secondary terms in arithmetic statistics

Abstract

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field ${\mathbb{F}}_q(t)$ of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of discriminant equal to $q^{2N}$ is $c_1{q}^{2N}-c_2^i{q}^{5N/3}+O_{\epsilon }\left(q^{(3/2+\epsilon )N}\right)$, where $c_1$ and $c_2^i$ are explicit constants and $c_2^i$ only depends on $N(\mathrm{mod}3)$. This builds on the work of Bhargava-Shankar-Tsimerman and Taniguchi-Thorne, who proved the existence of a secondary term for the count of cubic extensions of $\mathbb{Q}$ with bounded discriminant. Our approach uses a parametrization of Miranda and Casnati-Ekedahl, which can be seen as a geometric version of the classical parametrization by binary cubic forms used by Davenport-Heilbronn. This allows us to count and sieve for smooth curves embedded in Hirzebruch surfaces, in the same spirit as Zhao and Gunther.

65 pages. This was submitted (with minor formatting changes) to Harvard as the author's PhD thesis on April 3, 2025