Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit Construction of Maass Wave Forms and Their Petersson Inner Products

Daichi Tanaka

TL;DR

This work addresses the explicit construction of Maass wave cusp forms attached to Hecke characters on real quadratic fields and the explicit computation of their Petersson inner products. It develops an approach based on Hecke L-functions and Mellin-type techniques to build automorphic Maass cusp forms $\Theta_{\psi}$ from primitive Hecke characters of type $(\varepsilon,\varepsilon,\frac{\nu}{i},-\frac{\nu}{i})$, handling both parities and including the $\nu=0$ case. The paper then derives an explicit formula for $\left<\Theta_{\psi},\Theta_{\psi}\right>$ in terms of $\mathrm{Res}_{s=1}\zeta_{F}(s)$ and $L(1,\psi(\overline{\psi}\circ\sigma))$ with concrete constants, and provides dihedral-field examples expressing these inner products via regulators and zeta-values. Through dihedral Hilbert class-field constructions, it connects inner products to arithmetic invariants and Stark-type results, illustrating the arithmetic content and potential applications to $L$-values and norm relations.

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 representation.

Explicit Construction of Maass Wave Forms and Their Petersson Inner Products

TL;DR

This work addresses the explicit construction of Maass wave cusp forms attached to Hecke characters on real quadratic fields and the explicit computation of their Petersson inner products. It develops an approach based on Hecke L-functions and Mellin-type techniques to build automorphic Maass cusp forms from primitive Hecke characters of type , handling both parities and including the case. The paper then derives an explicit formula for in terms of and with concrete constants, and provides dihedral-field examples expressing these inner products via regulators and zeta-values. Through dihedral Hilbert class-field constructions, it connects inner products to arithmetic invariants and Stark-type results, illustrating the arithmetic content and potential applications to -values and norm relations.

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 representation.
Paper Structure (14 sections, 19 theorems, 135 equations)

This paper contains 14 sections, 19 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Hecke characters and Hecke $L$-functions
  3. Hecke characters
  4. Hecke $L$-functions
  5. Construction of Maass wave forms
  6. Proof for the case $\epsilon=0$
  7. Proof for the case $\epsilon=1$
  8. Computation of Petersson inner product
  9. Proof of Theorem \ref{['theorem;explicit computation of Petersson inner product']}
  10. Examples
  11. The dihedral case in general
  12. The case when $F=\mathbb{Q}(\sqrt{229})$
  13. The case when $F=\mathbb{Q}(\sqrt{445})$
  14. The case when $F=\mathbb{Q}(\sqrt{401})$

Key Result

Theorem 1.1

Let $F$ be a real quadratic field and let $\psi$ be a primitive Hecke character modulo $\mathfrak{f}$ on $F$. We denote by $D$ the discriminant of $F$. We assume that $\psi$ is of the type $(\epsilon,\epsilon,\frac{\nu}{i},-\frac{\nu}{i})$ and not of the form $\mathbb{N}_{F/\mathbb{Q}}\circ\chi$ for

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 34 more