A unified finiteness theorem for curves

TL;DR

This paper proves a finiteness theorem for Galois-invariant point sets on algebraic curves with controlled reduction, extending classical results like Faltings' theorem.

Contribution

It generalizes existing finiteness results by considering Galois-invariant sets with controlled reduction behavior on algebraic curves.

Findings

Finiteness of Galois-invariant point sets under automorphisms

Generalization of classical finiteness theorems

Breaks into finitely many orbits under automorphism group

Abstract

We study the arithmetic of Galois-invariant sets of points on algebraic curves with controlled reduction behavior. Let $C$ be a smooth projective curve with a smooth proper model $\mathcal{C}$ over $\mathcal{O}_{K,S}$. We define $\Omega_n$ as the set of $n$-element subsets of $C(\overline{K})$ that are invariant under $\text{Gal}(\overline{K}/K)$ and such that no two points in the set become identified modulo any prime $\mathfrak{p} \notin S$. Our main result establishes that $\Omega_n$ breaks into finitely many orbits under the action of $\text{Aut}_{\mathcal{O}_{K,S}}(\mathcal{C})$, generalizing finiteness theorems of Birch--Merriman, Siegel, and Faltings.