A Note on Divisible Points of Curves

TL;DR

This paper investigates points on algebraic curves within tori where non-trivial powers also lie on the curve, connecting to deep conjectures and employing transcendence and o-minimal methods.

Contribution

It provides partial answers to Levin's question about divisible points on curves in tori, linking to Zilber's Conjecture and introducing new methodological approaches.

Findings

Partial characterization of divisible points on curves in tori.

Connections established between algebraic geometry and transcendence theory.

Methodological advancements using o-minimal structures.

Abstract

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our results have connections to Zilber's Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.