Special values of anticyclotomic \(L\)-functions.
type: Item
instance of: scholarly article
title: Special values of anticyclotomic \(L\)-functions.
zbMATH Open document ID: 1065.11048
DOI: 10.1215/S0012-7094-03-11622-1
published in: Duke Mathematical Journal
publication date: 2003-01-01T00:00:00Z
review text: The author extends his earlier results and methods [\textit{V. Vatsal}, Invent. Math. 148, No.~1, 1--46 (2002; Zbl 1119.11035)] on nonvanishing of \(L\)-functions associated to modular forms in the anticyclotomic tower of conductor \(p^\infty\) over an imaginary quadratic field to the case where the sign in the functional equation is \(-1\) (Theorem 1.4). In that case he shows, in particular, that derivatives of the \(L\)-functions are generically nonzero at the center. The main ingredient in the proof is a new ``Johnowitz congruence'' that relates the nontriviality of a special value modular curve. Applications are given to the Iwasawa \(\mu\)-invariant of the \(p\)-adic \(L\)-functions of Bartolini and Darmon (Theorem 1.1).
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 1911547
zbMATH Keywords: modular form
zbMATH Keywords: anticyclotomic \(L\)-function
zbMATH Keywords: Heegner point
zbMATH Keywords: \(\mu\)-invariant
zbMATH Keywords: Galois representation
author: Vinayak Vatsal
MaRDI profile type: Publication
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full work available at URL: s0012-7094-03-11622-1
OpenAlex ID: W2090906809
description: scientific article; zbMATH DE number 1911547