The first eigenvalue problem and tensor products of zeta functions
type: Item
instance of: scholarly article
title: The first eigenvalue problem and tensor products of zeta functions
zbMATH Open document ID: 1065.11035
author: Shin-ya Koyama
published in: Proceedings of the Japan Academy. Series A
publication date: 2005-02-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1116442239
review text: The paper under review is concerned with Selberg's eigenvalue conjecture for the lowest positive eigenvalue \(\lambda_1\) of the Laplacian on a congruence subgroup of \(\text{SL}_2(\mathbb{Z})\) or \(\text{SL}_2({\mathcal O}_\alpha)\), where \({\mathcal O}_\alpha\) is the ring of integers of an imaginary quadratic number field of discriminant \(d<0\). The author introduces the following assumption on the integer \(n\geq 2\) and a cuspidal automorphic representation \(\pi\) of \(\text{GL}_2(\mathbb{A}_k)\) with a number field \(k\): There is a cuspidal automorphic representation \(\pi_n=\text{Sym}^{n-1}(\pi)\) of \(\text{GL}_n(\mathbb{A}_k)\) whose \(L\)-function is equal to \(L(s,\text{Sym}^{n-1}(\pi))\). This assumption is known to hold for \(n\geq 5\). If this assumption is true for all \(n\) then Selberg's conjecture holds true. Moreover, if this assumption holds for some integer \(n\geq 2\) the author gives lower bounds for \(\lambda_1\) depending on \(n\) which give the best known lower bounds in the two-dimensional case (for \(n=4\) or 5, respectively). For \(n=5\) this bound yields in the three-dimensional case (i.e. for congruence subgroups of \(\text{SL}_2({\mathcal O}_d)): \lambda_1\geq \frac{160}{169}=0.946\dots\) which is remarkably close to Selberg's conjecture \(\lambda_1\geq 1\). In the last section the author comments on the general principle that tensor products of zeta-functions will be useful for solving problems of the type of the Riemann hypothesis, and he introduces a possibly useful candidate.
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 2138332
zbMATH Keywords: Selberg conjecture
zbMATH Keywords: symmetric power \(L\)-function
zbMATH Keywords: tensor product of zeta-functions
MaRDI profile type: Publication
OpenAlex ID: W2092809250
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DOI: 10.3792/PJAA.80.35
description: scientific article; zbMATH DE number 2138332