Enumerating odd-degree hyperelliptic curves and abelian surfaces over $${\mathbb {P}}^1$$

Abstract Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature lower order terms. We consider arithmetic aspect inertia stack an algebraic over finite fields to partially answer this question. Subsequently, we acquire new sharp enumerations quasi-admissible odd-degree hyperelliptic curves $${\mathbb {F}}_q(t)$$ F q ( t ) ordered by discriminant height.