Electronic ISBN:  9781470425012 
Product Code:  MEMO/237/1117.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 237; 2015; 64 ppMSC: Primary 11;
The aim of this article is to give a complete account of the EichlerBrandt 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 Heckemodule 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 squarefree levels is also examined. These Heckeequivariant maps lead to a nice description of the subspace generated by the authors' theta series, and thereby contributes to the socalled 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 Shintanitype correspondence between Drinfeld type forms and metaplectic forms. 
Table of Contents

Chapters

1. Introduction

2. Brandt matrices and definite Shimura curves

3. The basis problem for Drinfeld type automorphic forms

4. Metaplectic forms and Shintanitype correspondence

5. Trace formula of Brandt matrices


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The aim of this article is to give a complete account of the EichlerBrandt 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 Heckemodule 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 squarefree levels is also examined. These Heckeequivariant maps lead to a nice description of the subspace generated by the authors' theta series, and thereby contributes to the socalled 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 Shintanitype correspondence between Drinfeld type forms and metaplectic forms.

Chapters

1. Introduction

2. Brandt matrices and definite Shimura curves

3. The basis problem for Drinfeld type automorphic forms

4. Metaplectic forms and Shintanitype correspondence

5. Trace formula of Brandt matrices