# A Schanuel Property for $j$

## TL;DR

This paper develops a model-theoretic framework for the modular j function and its derivatives, establishing a generic transcendence property that supports the Ax-Schanuel theorem within this setting.

## Contribution

It introduces the concept of j-fields as a new model-theoretic structure for the j function, extending ideas from exponential fields to modular functions.

## Key findings

- Proves a generic transcendence property for the j function.
- Provides a model-theoretic setting for the Ax-Schanuel theorem for j.
- Establishes foundational results for the model theory of modular functions.

## Abstract

I give a model-theoretic setting for the modular $j$ function and its derivatives. These structures, here called $j$-fields, provide an adequate setting for interpreting the Ax-Schanuel theorem for $j$ (Pila-Tsimerman 2015). Following the ideas of M. Bays, J. Kirby and A.J. Wilkie for exponential fields, I prove a generic transcendence property for the $j$ function.