Rankin-Selberg \(L\)-functions in the level aspect.
type: Item
instance of: scholarly article
title: Rankin-Selberg \(L\)-functions in the level aspect.
zbMATH Open document ID: 1035.11018
DOI: 10.1215/S0012-7094-02-11416-1
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1087575359
review text: A well-known problem in the theory of automorphic forms is to study the size of \(L\)-values, especially on the critical line. The ``convexity'' bound is obtained by combining the absolute convergence to the right with the functional equation and then employing the Phragmen-Lindelöf method. One seeks estimates that break this convexity bound. For example, \textit{W. Duke, J. Friedlander} and \textit{H. Iwaniec} [Invent. Math. 115, 209--217 (1994; Zbl 0791.11049)] obtained such an estimate for the second moment of the quadratic twists of the family of primitive cuspidal newforms on \(\Gamma_0(q)\) with trivial nebentypus and weight \(k\) (denoted \(S_k^*(q)\)). In this paper the authors give asymptotic estimates for the modulus of the Rankin-Selberg \(L\)-functions (and their derivatives) \(L^{(j)}(f\otimes g,1/2+it)\). Here \(g\) is a fixed modular form on \(\Gamma_1(D)\), \(D\) square-free, which is either a primitive cuspidal holomorphic newform of integral weight or a non-exceptional Maass form of weight zero, and the estimate obtained is valid for all \(f\in S_k^*(q)\) with \(k\geq2\) even and \((q,D)=1\). As a consequence the authors give an unconditional improvement on the trivial bound for the problem of distinguishing primitive cusp forms by their first Fourier coefficients. They also present asymptotic results for the second moments of the Rankin-Selberg values for \(q\) prime and \(k<12\), and use this to prove various nonvanishing results for the critical values \(L(f\otimes g,1/2)\) (resp.\ \(L'(f\otimes g,1/2)\) when the vanishing is to odd order). In addition, they give a linear independence result for certain isotypic components of the images of Heegner divisors under the first Hecke operators. The authors' main result follows from the amplification method of Friedlander and Iwaniec and their previous work [Invent. Math. 142, 95--151 (2000; Zbl 1054.11026)]. To take care of terms arising from the level and nebentypus of \(g\) the authors employ the th…
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1820921
zbMATH Keywords: moments
zbMATH Keywords: Rankin-Selberg convolution
zbMATH Keywords: level aspect
zbMATH Keywords: convexity-breaking
Wikidata QID: Q57268237
author: E. Kowalski
author: Philippe Michel
author: Jeffrey M. Vanderkam
MaRDI profile type: Publication
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OpenAlex ID: W2030914290
description: scientific article; zbMATH DE number 1820921