A unified finiteness theorem for curves

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 C over OK,S. We define Ωₙ as the set of n-element subsets of C(K̄) that are invariant under Gal(K̄/K) and such that no two points in the set become identified modulo any prime p ∉ S. Our main result establishes that Ωₙ breaks into finitely many orbits under the action of Aut_OK,S(C), generalizing finiteness theorems of Birch--Merriman, Siegel, and Faltings.