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