The geometry of secondary terms in arithmetic statistics

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 $c_1 q^{2N} - c_2^{i} q^{5N/3} + O_{\varepsilon}\left(q^{(3/2+\varepsilon)N}\right)$, where $c_1$ and $c_2^{i}$ are explicit constants and $c_2^{i}$ only depends on $N\pmod{3}$. 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.