Characterizing algebraic curves with infinitely many integral points

TL;DR

This paper provides necessary and sufficient conditions for algebraic curves over number fields to have infinitely many S-integral points, extending classical results on finiteness of integral points.

Contribution

It characterizes algebraic curves with infinitely many S-integral points, refining Siegel's theorem with precise conditions.

Findings

Identifies conditions for infinite S-integral points on curves

Extends classical finiteness results to new cases

Provides a complete characterization of such curves

Abstract

A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have infinitely many S-integral points.