Automorphic pseudodifferential analysis and higher level Weyl calculi
type: Item
instance of: scholarly article
title: Automorphic pseudodifferential analysis and higher level Weyl calculi
zbMATH Open document ID: 1018.11018
author: André Unterberger
published in: Progress in Mathematics
publication date: 2003-01-28T00:00:00Z
review text: The group \(\text{SL} (2,\mathbb R)\) acts on the Poincaré upper half plane \(\mathcal H\) by linear fractional transformations. Given a discrete subgroup \(\Gamma\) of \(\text{SL} (2,\mathbb R)\), automorphic functions are \(\Gamma\)-invariant functions on \(\mathcal H\). A nonholomorphic modular form is an automorphic function on \(\mathcal H\) that is a generalized eigenfunction of the Laplace-Beltrami operator \(\Delta\). Automorphic functions can also be regarded as objects defined on \(\mathbb R^2\), rather than on \(\mathcal H\), which are invariant under the linear action of \(\Gamma\). Such an interpretation requires the use of distributions instead of functions, so that automorphic functions on \(\mathcal H\) correspond to automorphic distributions on \(\mathbb R^2\). This monograph studies a class of pseudodifferential operators in one variable whose Weyl symbols are automorphic distributions on \(\mathbb R^2\), that is, distributions invariant under the linear action of \(\Gamma\). By using the spectral theory of the Euler operator on \(L^2 (\Gamma \backslash \mathbb R^2)\), it can be shown that automorphic distributions are linear superpositions of the Eisenstein distributions and the cusp distributions. The main goal of this book is to construct a multiplication table for the associated operators. The space of automorphic functions on the upper half plane can naturally be turned into a noncommutative algebra by adopting the quantum approach. This is achieved by associating operators to automorphic functions and considering the composition of those operators. The resulting composition formulas reveal various important objects in the theory of nonholomorphic modular forms such as zeta functions, Hecke operators, \(L\)-functions, and convolution of \(L\)-functions. One advantage of using distributions is that the algebraic structure on the space of automorphic distributions can be defined solely in terms of Weyl calculus of operators. The main result in t…
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 11F12
zbMATH DE Number: 1860220
zbMATH Keywords: automorphic forms
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Weyl calculus
zbMATH Keywords: automorphic distributions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1860220