Estimating additive character sums for Fuchsian groups
type: Item
instance of: scholarly article
title: Estimating additive character sums for Fuchsian groups
zbMATH Open document ID: 1047.11049
DOI: 10.1023/A:1026255414488
author: Dorian Goldfeld
author: Cormac O'Sullivan
published in: The Ramanujan Journal
publication date: 2004-01-06T00:00:00Z
review text: Some forty years ago, in connection with the (Eichler) cohomology of automorphic forms, \textit{M. Eichler} [Acta Arith. 11, 169--180 (1965; Zbl 0148.32503)] introduced (what later came to be called) the ``generalized Poincaré (in particular, Eistenstein) series''. Following this lead, Lehner and, somewhat later, the reviewer made use of Eichler's generalized Poincaré series in proving versions of the Eichler cohomology theorem. In these earlier applications the series in question are holomorphic in \(H\), the upper half-plane. More recently, Goldfeld and his school initiated a natural new direction in the study of such series, considering a nonholomorhpic (real-analytic in \(H\)) version analogous to the nonanalytic Poincaré series introduced by Hecke in 1921. This path-breaking work of Hecke shifted the focus from the \(z\) variable (\(z\) in \(H\)), with respect to which Hecke's series behave like modular forms on a subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb{Z})\), to the new \(s\) variable, with \(s\) in a right half-plane. These series are holomorphic in \(s\) for \(\text{Re\,}s\) sufficiently large; Hecke succeeded in continuing them analytically into an open half-plane containing \(s= 0\) and, consequently, in constructing nontrivial modular forms of weights 1 and 2 on \(\Gamma\). In the paper under review the authors continue the study of nonanalytic generalized Poincaré series and apply their results to the estimation of character sums on finitely generated Fuchsian groups \(\Gamma\) of the first kind. The characters in question are mappings from \(\Gamma\) into \(\mathbb{C}\). The presence of these (weight 0) characters distinguish the generalized Poincaré series (holomorphic in \(z\) or not) from the usual Poincaré series, as they replace the multiplicative characters (i.e., weight 0 multiplier systems) that occur in the latter. This replacement entails a loss of group invariance, with respect to \(\Gamma\), which is, however, supplanted by a more compl…
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 2022031
zbMATH Keywords: Eisenstein series
zbMATH Keywords: cusp forms
zbMATH Keywords: modular symbols
zbMATH Keywords: character sums
zbMATH Keywords: Kloosterman sums
MaRDI profile type: Publication
full work available at URL: a:1026255414488
OpenAlex ID: W4239370500
description: scientific article; zbMATH DE number 2022031