Higher order Maass forms

TL;DR

This paper investigates the size and structure of higher order Maass forms and holomorphic forms for certain groups, establishing maximality results and introducing generalized weights for functions on the universal covering group.

Contribution

It introduces the concept of generalized weight for functions on the universal covering group and proves maximality of the size of higher order Maass form spaces with even generalized weight.

Findings

Spaces of higher order Maass forms are as large as algebraic constraints allow.

Analogous maximality results are shown for spaces of holomorphic forms.

The concept of generalized weight is introduced for functions on the universal covering group.

Abstract

We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as algebraic constrictions allow. We show the analogous statement for the spaces of holomorphic forms. For functions on the universal covering group of PSL(2,R) we introduce the concept of generalized weight. For the resulting spaces of higher order Maass forms with even generalized weight we show that the size is maximal.