Estimating additive character sums for Fuchsian groups
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 complicated, three-term analogue for the generalized series. The application to estimating character sums is a striking development, which appears to have the potential for extension to sums over \(\Gamma\) of nonzero-weight additive cocycles on \(\Gamma\). The estimation requires extensive analysis of the generalized Poincaré series, including especially analytic continuation in the variable \(s\). The authors emphasize the relatively difficult case of noncuspidal additive characters on \(\Gamma\). (Cuspidal characters are those that vanish on all parabolic generators of \(\Gamma\)).
scientific article; zbMATH DE number 2022031