Rational period functions of the modular group. (Appendix by Georges Grinstein)
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.
scientific article; zbMATH DE number 3582238