A unified finiteness theorem for curves

TL;DR

The paper proves a unified finiteness theorem for Galois-invariant $n$-point subsets on a smooth projective curve with controlled reduction, showing that the orbits under ${\rm Aut}_{\mathcal{O}_{K,S}}(\mathcal{C})$ of $\Omega_{n,\overline{K}}(C;S)$ are finite. It analyzes the problem by genus, first establishing finiteness for $K$-points with a standard model using classical results (Birch–Merriman, Siegel, Faltings) and then extends to $\Omega_{n,\overline{K}}(C;S)$ via a descent argument: pass to a large field $H$ where all points become rational, control fibers with $H^1(G,M_A)$, and descend back to $K$. The work recovers and unifies several classical finiteness theorems (class group finiteness, Birch–Merriman, Siegel, Faltings) as special cases and provides a framework to count horizontal divisors of degree $n$ up to automorphisms on arithmetic surfaces. The methods combine reduction maps, binary form techniques, and non-abelian Galois cohomology to achieve finiteness results with potential applications to arithmetic geometry and Diophantine problems involving point configurations on curves.

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 $Ω_n$ 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 $Ω_n$ 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.

Theorems & Definitions (32)

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