A Unified Finiteness Theorem For Curves Over Function Fields

TL;DR

This paper generalizes a finiteness theorem for curves over number fields to the setting of function fields over finite fields, establishing finiteness of certain divisors under specified conditions.

Contribution

It extends the unified finiteness theorem to function fields, providing a broader understanding of divisors over curves in this setting.

Findings

Finiteness of horizontal divisors of fixed degree over function fields.

Extension of previous results from number fields to function fields.

Finiteness holds up to automorphisms and Frobenius in the isotrivial case.

Abstract

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 $\pi: S \rightarrow C$ 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 \'etale over the base $C$, up to the action of the automorphism group and Frobenius (in the isotrivial case).