The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points
type: Item
instance of: scholarly article
title: The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points
zbMATH Open document ID: 1068.11033
published in: Annals of Mathematics. Second Series
publication date: 2005-04-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.annm/1105737692
review text: The author considers Rankin-Selberg \(L\)-functions \(L(f\otimes g,s)\), where \(f\) and \(g\) are primitive cusp forms with \(g\) fixed and holomorphic of weight \(\geq 1\). Let \(f\) and \(g\) be of level \(q\), \(D\) and of Nebentypus \(\chi _f\), \(\chi _g\), respectively, and suppose that \(\chi _f\chi _g\) is not trivial. The main result of the paper is the estimate \(L^{(j)}(f\otimes g,s) \ll q ^{1/2-1/1057}\), where \(\Re s =1/2\) and the implied constant depends on \(j\), \(s\), \(D\) and of the weight or spectral parameter of \(f\) depending on whether \(f\) is holomorphic or real analytic. The point here is the subconvexity of the estimate as compared with the convexity bound \(\ll q^{1/2+\varepsilon }\). This is a generalization of an earlier result due to \textit{E. Kowalski}, \textit{P. Michel} and \textit{J. Vanderkam} [Duke Math. J. 114, 123--191 (2002; Zbl 1035.11018)]. The proof is very complicated and ingenious, being based on the amplification technique. The equidistribution of certain Heegner points is discussed as an application.
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 2151002
zbMATH Keywords: Rankin-Selberg \(L\)-function
zbMATH Keywords: subconvexity
zbMATH Keywords: Heegner points
author: Philippe Michel
Wikidata QID: Q95774695
MaRDI profile type: Publication
OpenAlex ID: W3023694340
DOI: 10.4007/ANNALS.2004.160.185
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description: scientific article; zbMATH DE number 2151002