\(ABC\) implies no ``Siegel zeros'' for \(L\)-functions of characters with negative discriminant
type: Item
instance of: scholarly article
title: \(ABC\) implies no ``Siegel zeros'' for \(L\)-functions of characters with negative discriminant
zbMATH Open document ID: 0967.11033
author: Andrew Granville
author: Harold M. Stark
published in: Inventiones Mathematicae
publication date: 2001-07-12T00:00:00Z
review text: \textit{P. Vojta} [Diophantine approximations and value distribution theory. Lecture Notes in Mathematics 1239, Springer-Verlag (1987; Zbl 0609.14011)] has given a version of the \(abc\)-conjecture for number fields, with full uniformity with respect to the field. The present paper shows how this conjecture implies the existence of a positive constant \(c\) such that \(L(s, \chi)\neq 0\) for \(s> 1- c/\log d\), when \(\chi\) is the character corresponding to the imaginary quadratic field \(\mathbb{Q}(\sqrt{-d})\). The proof goes via an estimate for the class number, which enters through a bound for the height of the \(j\)-invariant, evaluated at either \(\tau= (-1+ \sqrt{-d})/2\) or \(\sqrt{-d}/2\) as appropriate. The \(j\)-function appears in the Diophantine equation \[ j(\tau)= \gamma_2(\tau)^3= \gamma_3(\tau)^3+ 1728 \] where \(\gamma_2(\tau)\) and \(\gamma_3(\tau)\) lie in a certain ray class field. The paper also investigates other Diophantine equations over class fields, which arise from other modular relations. This produces new (conjecturally) best possible examples in the uniform \(abc\)-conjecture.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 1443585
zbMATH Keywords: class field
zbMATH Keywords: Dirichlet \(L\)-function
zbMATH Keywords: real zero
zbMATH Keywords: exceptional zero
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: \(abc\)-conjecture
MaRDI profile type: Publication
full work available at URL: s002229900036
OpenAlex ID: W2058408537
Wikidata QID: Q55968421
DOI: 10.1007/S002229900036
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description: scientific article; zbMATH DE number 1443585