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2020 | Buch

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On the Theory of Maass Wave Forms

verfasst von: Dr. Tobias Mühlenbruch, Wissam Raji

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook provides a rigorous analytical treatment of the theory of Maass wave forms. Readers will find this unified presentation invaluable, as it treats Maass wave forms as the central area of interest. Subjects at the cutting edge of research are explored in depth, such as Maass wave forms of real weight and the cohomology attached to Maass wave forms and transfer operators. Because Maass wave forms are given a deep exploration, this book offers an indispensable resource for those entering the field.
Early chapters present a brief introduction to the theory of classical modular forms, with an emphasis on objects and results necessary to fully understand later material. Chapters 4 and 5 contain the book’s main focus: L-functions and period functions associated with families of Maass wave forms. Other topics include Maass wave forms of real weight, Maass cusp forms, and weak harmonic Maass wave forms. Engaging exercises appear throughout the book, with solutions available online.
On the Theory of Maass Wave Forms is ideal for graduate students and researchers entering the area. Readers in mathematical physics and other related disciplines will find this a useful reference as well. Knowledge of complex analysis, real analysis, and abstract algebra is required.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. A Brief Introduction to Modular Forms

Abstract

In this chapter, we give an overview on the theory of “classical” modular forms. What we mean by classical here are holomorphic modular forms on the full modular group or on its congruence subgroups. We give the definition of modular forms for real weights and give properties of the multiplier system. A few explicit examples of modular forms (Eisenstein series, discriminant function, holomorphic Poincaré series, and theta series) are also presented. Then, we introduce Hecke operators and L-functions and give their properties that will be needed in later chapters.

Tobias Mühlenbruch, Wissam Raji

Chapter 2. Period Polynomials

Abstract

In this chapter, we discuss period polynomials that emerge from Eichler integrals on the full modular group. Period polynomials and their coefficients have significant impact on the theory of modular forms. In fact, the coefficients of these polynomials are certain values of L-functions. Moreover, by defining the cohomology of period polynomials, one can show that there is an isomorphism between two copies of the space of modular forms and the cohomology group. This is known as the Eichler-Shimura isomorphism. We give all the basic properties of period polynomials and prove the Eichler-Shimura isomorphism theorem. We continue by describing the action of Hecke operators on period polynomials, following an approach by Choie and Zagier. We omit some of the proofs due to the abundance of references in the literature or, in many cases, because we prove their analogues in the Maass forms case.

Tobias Mühlenbruch, Wissam Raji

Chapter 3. Maass Wave Forms of Real Weight

Abstract

In this chapter, we introduce Maass wave/cusp forms of real weight on subgroups of finite index in the full modular group with compatible multiplier systems. After introducing multiplier systems and the unitary automorphic factor and discussing the hyperbolic Laplace operator and Maass operators, we define Maass waveforms. We then use Hecke operators in the special case where the Maass waveform is of weight 0 and trivial multiplier system on the full group. We conclude the chapter by mentioning some spectral theory for the Laplace operator and its Friedrichs extension and discuss briefly Selberg’s conjecture.

Tobias Mühlenbruch, Wissam Raji

Chapter 4. Families of Maass Cusp Forms, L-Series, and Eichler Integrals

Abstract

We saw in the previous chapter that Maass operators E ± k map Maass cusp forms of weight k to Maass cusp forms of weight k ± 2. Can we exploit this result and use Maass operators to get a whole family of Maass cusp forms by starting with a “reference” Maass waveform of weight k and generate all possible images by repeatedly applying one of the Maass operators?

Tobias Mühlenbruch, Wissam Raji

Chapter 5. Period Functions

Abstract

We introduce period functions and give several relations to (families of) Maass cusp forms. We present some remarks about Zagier’s cohomology theorem that generalizes the Eichler cohomology theorem. We conclude this chapter with a discussion of Hecke operators acting on period functions.

Tobias Mühlenbruch, Wissam Raji

Chapter 6. Continued Fractions and the Transfer Operator Approach

Abstract

The transfer operator approach is a concept first introduced by David Ruelle, a Belgian-French mathematical physicist, who worked on statistical physics and dynamical systems. He introduced the transfer operator a means to study dynamical zeta functions. A good introduction to chaotic systems, motivating and introducing dynamical zeta functions (in a general setting) is the web book. Our goal in this chapter is more specific. We focus solely on variants of the Artin billiard and the Mayer’s transfer operator approach. Surprisingly, the period functions, as defined in §5.1 of the previous chapter, play a significant role in Mayer’s transfer operator theory.

Tobias Mühlenbruch, Wissam Raji

Chapter 7. Weak Harmonic Maass Wave Forms

Abstract

In this chapter, we introduce the newly emerging concept of weakly harmonic Maass wave forms.

Tobias Mühlenbruch, Wissam Raji

Backmatter

Titel: On the Theory of Maass Wave Forms
verfasst von: Dr. Tobias Mühlenbruch
Wissam Raji
Copyright-Jahr: 2020
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-40475-8
Print ISBN: 978-3-030-40477-2
DOI: https://doi.org/10.1007/978-3-030-40475-8

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