Generic Solutions of Equations Involving the Modular $j$-function

TL;DR

This paper explores the existence of solutions to equations involving the modular j-function, linking it to deep conjectures in number theory and algebraic geometry, and providing conditions for unconditional results.

Contribution

It reduces the problem of finding solutions to modular j-equations to Zariski density, assuming conjectures, and offers unconditional results under certain field conditions.

Findings

Conditional reduction to Zariski dense solutions

Unconditional results under specific field assumptions

Connections to modular Schanuel and Zilber-Pink conjectures

Abstract

Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular $j$ function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain unconditional versions of this result.