A Unified Finiteness Theorem For Curves Over Function Fields

Motivated by the analogy between number fields and function fields, this paper extends the main result of \cite{janbazi2025unified} to the function field setting. Let $C$ be a smooth affine curve over a finite field, and let $\pi :S\to C$ be a smooth, proper model of a curve over $C$. Then, for any fixed integer $n\in \mathbb{N}$, there are only finitely many horizontal divisors of degree $n$ that are étale over the base $C$, up to the action of the automorphism group and Frobenius (in the isotrivial case).