On the counting function of cubic function fields

Abstract: We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches the best-known result, due to Bhargava, Taniguchi and Thorne, over $\mathbb{Q}$. Furthermore, we obtain estimates for the refined counting function, where one specifies the splitting behaviour of finitely many primes. Also in this case, our error term matches what is known for number fields. However, in the function field setting, the secondary term becomes more difficult to write down explicitly. Our proof uses geometry of numbers methods, which are especially effective for function fields. In particular, we obtain an exact formula for the number of orbits of cubic forms with fixed absolute discriminant. Moreover, by studying the one-level density of a family of Artin $L$-functions associated to these cubic fields, we prove an unconditional lower bound on the error term in the estimate for the refined counting function. This generalises a conditional result over $\mathbb{Q}$, due to Cho, Fiorilli, Lee and S\"odergren.