A proof of the Mahler-Manin conjecture
type: Item
instance of: scholarly article
title: A proof of the Mahler-Manin conjecture
zbMATH Open document ID: 0853.11059
author: Katia Barré-Sirieix
author: Guy Diaz
author: Georges Philibert
author: François Gramain
published in: Inventiones Mathematicae
publication date: 1997-01-09T00:00:00Z
review text: This paper answers the question of the transcendence values for the Fourier expansion at infinity of the modular invariant \(j\) at algebraic points. More precisely, \(j(q)\) is proved to be transcendental over the field of rationals (respectively over the field of \(p\)-adic rationals) for any non-zero algebraic point \(q\) of the unit disk of the complex plane (respectively of the completion of the field of \(p\)-adic rationals). This result conjectured by Mahler in the complex case and by Manin in the \(p\)-adic case has numerous applications in the theory of elliptic curves and \(p\)-adic \(L\) functions. The proof inspired by Mahler's method is based on sharp estimates of the coefficients of the modular polynomials. Let us note that a characteristic \(p\) analogue of this result has been proved by \textit{J. F. Voloch} [J. Number Theory 58, 55-59 (1996; Zbl 0853.11061)], see also \textit{D. S. Thakur}'s proof based on the theory of automata [J. Number Theory 58, 60-63 (1996; Zbl 0853.11060)].
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 866083
zbMATH Keywords: Mahler-Manin conjecture
zbMATH Keywords: modular invariant
zbMATH Keywords: transcendence values for the Fourier expansion at infinity
zbMATH Keywords: modular polynomials
Wikidata QID: Q122884624
reviewed by: Valérie Berthé
MaRDI profile type: Publication
full work available at URL: s002220050044
OpenAlex ID: W2002551472
DOI: 10.1007/S002220050044
description: scientific article; zbMATH DE number 866083