Rational period functions of the modular group. (Appendix by Georges Grinstein)
type: Item
instance of: scholarly article
title: Rational period functions of the modular group. (Appendix by Georges Grinstein)
zbMATH Open document ID: 0374.10014
DOI: 10.1215/S0012-7094-78-04504-0
published in: Duke Mathematical Journal
publication date: 1978-01-01T00:00:00Z
review text: Suppose \(G\) is a Fuchsian group and \(C(x)\) is the field of rational functions in \(x\). For \(k\in\mathbb Z\), \(f \in C(x)\) define \(f\vert_kg(x) = g'(x)^k f(g(x))\). This makes \(C(x)\) into a \(G\)-module, \(A_k\). To study this module one looks at the (parabolic) cohomology groups, which are analogous to Eichler cohomology. The author does this when \(G = \Gamma\), the modular group and \(k> 0\), \(k\) odd. Classically, this is motivated by the transformation behaviour of the Hecke function \(G\). The author constructs for each such \(k\) a parabolic 1-cocycle. In a previous work he examined the splitting of such cocycles in the corresponding \(G\)-module of functions analytic on the upper half-plane; one of the objects of this paper is to study the Mellin transform of such functions. Moreover, he studies the action of the Hecke algebra on the group of 1-cocycles and shows that it cannot be diagonalised.
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 3582238
zbMATH Keywords: rational period functions
zbMATH Keywords: modular group
zbMATH Keywords: parabolic 1-cocycle
zbMATH Keywords: Mellin transform
zbMATH Keywords: action of Hecke algebra
author: Marvin Knopp
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3582238