# Brandt Matrices and Theta Series over Global Function Fields

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*Chih-Yun Chuang; Ting-Fang Lee; Fu-Tsun Wei; Jing Yu*

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.

#### Table of Contents

# Table of Contents

## Brandt Matrices and Theta Series over Global Function Fields

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Brandt matrices and definite Shimura curves 714 free
- Chapter 3. The basis problem for Drinfeld type automorphic forms 1724
- Chapter 4. Metaplectic forms and Shintani-type correspondence 3340
- Chapter 5. Trace formula of Brandt matrices 5158
- Bibliography 6168
- Symbols 6370
- Back Cover Back Cover176