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.
Categories: math.NT math.AG