Brandt matrices and theta series over global function fields / Chih-Yun Chuang [and three others].

Format:

Book

Author/Creator:

Chuang, Chih-Yun, 1984-

Chuang, Chih-Yun, 1984- author.

Series:

Memoirs of the American Mathematical Society ; Volume 237, Number 1117.

Memoirs of the American Mathematical Society, 1947-6221 ; Volume 237, Number 1117

Language:

English

Subjects (All):

Matrices.

Quaternions.

Hecke algebras.

Series, Theta.

Physical Description:

1 online resource (69 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2015.

Language Note:

English

Summary:

The aim of this article is to give a complete account of the Eichler-Brandt theory over function fields and the basis problem for Drinfeld type automorphic forms. Given arbitrary function field k together with a fixed place \infty, the authors construct a family of theta series from the norm forms of "definite" quaternion algebras, and establish an explicit Hecke-module homomorphism from the Picard group of an associated definite Shimura curve to a space of Drinfeld type automorphic forms. The "compatibility" of these homomorphisms with different square-free levels is also examined. These Hecke-equivariant maps lead to a nice description of the subspace generated by the authors' theta series, and thereby contributes to the so-called basis problem. Restricting the norm forms to pure quaternions, the authors obtain another family of theta series which are automorphic functions on the metaplectic group, and this results in a Shintani-type correspondence between Drinfeld type forms and metaplectic forms.

Contents:

Cover

Title page

Chapter 1. Introduction

Chapter 2. Brandt matrices and definite Shimura curves

1. Basic setting

2. Definite quaternion algebra over function fields

3. Brandt matrices

4. Definite Shimura curves

4.1. Hecke correspondences

4.2. Gross height pairing

Chapter 3. The basis problem for Drinfeld type automorphic forms

1. Weil representation

1.1. Weil representation of \SL₂× ( )

1.2. Test functions from arithmetic data

2. Theta series

3. Drinfeld type automorphic forms and Hecke operators

3.1. Fourier coefficients of theta series

4. The Hecke module homomorphism Φ

4.1. Changing levels

5. Construction of Drinfeld type newforms

6. The basis problem

Chapter 4. Metaplectic forms and Shintani-type correspondence

1. Metaplectic forms

1.1. Metaplectic group

1.2. Weil representation and theta series from pure quaternions

1.3. Fourier coefficients of metaplectic theta series

2. Hecke operators and Shintani-type correspondence

3. Pure quaternions and Brandt matrices

Chapter 5. Trace formula of Brandt matrices

1. Optimal embeddings

1.1. Local optimal embeddings

2. Trace formula

Bibliography

Symbols

Back Cover.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-2501-7