Rational Period Functions and Parabolic Cohomology Original Research Article

In [K], M. Knopp defined a generalization of Eichler cohomology by considering rational functions as possible periods for the action, by way of the usual slash operator, of a Fuchsian group upon functions defined on the upper half-plane. This theory of rational period functions already enjoys a rich history. A significant step was made by A. Ash [A], who applied cohomological techniques in the setting of the finite index subgroups of the modular group to provide a classification of their rational period functions. Here we show that Ashʹs theorem, with appropriate adjustments, is valid for all finitely generated Fuchsian groups of the first kind with parabolic elements. Ashʹs proof relies heavily upon the Borel–Serre compactification forarithmeticgroups. We show that this compactification is valid in our wider setting and proceed to give a simplified version of the Ash proof. Applications to the classification of rational period functions of the Hecke groups are provided.