Explicit Construction of Maass Wave Forms and Their Petersson Inner Products

TL;DR

This paper explicitly constructs Maass wave cusp forms linked to Hecke characters over real quadratic fields, generalizing previous work and providing explicit Petersson inner products with illustrative examples.

Contribution

It introduces a method to explicitly construct Maass wave forms for arbitrary real quadratic fields, extending Maass's original work to broader cases.

Findings

Explicit construction of Maass wave forms for general real quadratic fields

Calculation of Petersson inner products for these forms

Examples involving dihedral Artin representations

Abstract

In this paper, we explicitly construct Maass wave cusp forms associated to Hecke characters on arbitrary real quadratic fields. This result is a generalization of Maass (1949), who constructed Maass wave cusp forms under the assumption that narrow class number is one. We also compute its Petersson inner product explicitly and give a few examples involving dihedral Artin representations.