Modular \(L\)-values of cubic level
type: Item
instance of: scholarly article
title: Modular \(L\)-values of cubic level
zbMATH Open document ID: 1286.11069
author: Andrew Knightly
author: Charles Li
published in: Pacific Journal of Mathematics
publication date: 2013-02-15T00:00:00Z
full work available at URL: p10.xhtml
review text: Let \(\chi\) be a fixed primitive Dirichlet character modulo \(D\) and \(N\) a squarefree integer coprime to \(D\). The paper studies certain averages of \(\chi\)-twisted \(L\)-values over (i) even Maass newforms on \(\Gamma_0(N^3)\) of weight \(k=0\) with trivial central character, or (ii) holomorphic newforms on \(\Gamma_0(N^3)\) of weight \(k > 2\) whose central character satisfies \(\omega'(-1)=(-1)^k\), as \(N \to \infty\). These results correspond to Theorem 5.4 and Theorem 4.1, respectively. In the Maass case there is even an exact formula. There are two ingredients in their approach. The global one is Jacquet's relative trace formula for \(\mathrm{GL}(2)\) with respect to its spherical subgroups \(N\) and \(M\), against suitable characters. Here \(N\) is the upper triangular unipotent subgroup and \(M\) is the diagonal torus \(\mathrm{diag}(\cdot, 1)\). Integration over \(N\) produces Fourier coefficients \(a_r(u)\), whereas a classical result by Hecke asserts that integration over \(M\) produces \(L\)-values. In order to exploit the relative trace formula, one chooses suitable test functions to project to some given spectrum \(A_k(\hat{\sigma}, \omega)\), then one computes the relevant terms in the trace formula. The local ingredient concerns the choice of local data (supercuspidal representations) at \(p\)-adic places. Such representations have explicit matrix coefficients (in terms of Kloosterman sums!) at least for \(\mathrm{GL}(2)\). The paper contains many details for the relevant explicit computations.
reviewed by: Wen-Wei Li
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 6136442
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: relative trace formula
zbMATH Keywords: supercuspidal representations
zbMATH Keywords: Maass forms
MaRDI profile type: Publication
OpenAlex ID: W2088210690
DOI: 10.2140/PJM.2012.260.527
description: scientific article; zbMATH DE number 6136442