Ax–Schanuel for the j-function

Jonathan Pila; Jacob Tsimerman

doi:10.1215/00127094-3620005

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.

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Jonathan Pila. Jacob Tsimerman. "Ax–Schanuel for the $j$ -function." Duke Math. J. 165 (13) 2587 - 2605, 15 September 2016. https://doi.org/10.1215/00127094-3620005

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Received: 21 January 2015; Revised: 28 September 2015; Published: 15 September 2016

First available in Project Euclid: 10 June 2016

zbMATH: 06650079

MathSciNet: MR3546969

Digital Object Identifier: 10.1215/00127094-3620005

Subjects:

Primary: 11C18

Secondary: 03C98

Keywords: Ax–Schanuel theorem , modular curve , number theory , Shimura varieties , transcendence

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