Roots of Modular Functions

TL;DR

This paper proves a bound on the level of roots of modular units, showing that if a modular unit's p-th root is also a modular unit, then the level increases by at most a factor of p.

Contribution

It establishes a new level bound for roots of modular units, advancing understanding of their algebraic structure.

Findings

If a modular unit has a p-th root that is also a modular unit, then the root's level is at most p times the original level.

The result constrains the possible levels of roots of modular units, impacting their classification.

Provides a theoretical bound relevant to the study of modular functions and their roots.

Abstract

Let $p$ be a prime. We prove that if a modular unit has a $p^{th}$ root that is again a modular unit then the level of that root is at most $p$ times the level of the original unit.