On a modular Fermat equation

TL;DR

This paper explores diophantine problems inspired by the analogy between multiplicative groups and modular curve powers, proving a special case of the Zilber-Pink conjecture for curves.

Contribution

It establishes a specific case of the Zilber-Pink conjecture related to modular curves, advancing understanding of unlikely intersections.

Findings

Proved a special case of the Zilber-Pink conjecture for curves.

Linked diophantine problems with modular curves and unlikely intersections.

Enhanced the theoretical framework connecting multiplicative groups and modular curves.

Abstract

We consider some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of "unlikely intersections." We prove a special case of the Zilber-Pink conjecture for curves.