Ax-Schanuel for the j-function

Abstract

In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations.

Comments Welcome!

Cited by in corpus (19)

• Applications of the hyperbolic Ax-Schanuel conjecture
• When any three solutions are independent
• Ax-Schanuel and strong minimality for the $j$-function
• Blurrings of the $j$-function
• Differential Existential Closedness for the $j$-function
• A Schanuel Property for $j$
• Weak Modular Zilber-Pink with Derivatives
• A closure operator respecting the modular $j$-function
• Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields
• Mixed Ax-Schanuel for the universal abelian varieties and some applications
• Solving Systems of Equations of Raising-to-Powers Type
• Generic Solutions of Equations Involving the Modular $j$-function
• Ax-Schanuel with derivatives for mixed period mappings
• Solutions of equations involving the modular $j$ function
• Algebraic Varieties and Automorphic Functions
• Nondefinability results for elliptic and modular functions
• Ax-Schanuel for $G{L}_n$
• Enlarged mixed Shimura varieties, bi-algebraic system and some Ax type transcendental results
• Multiplicative independence of modular functions