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# memo_has_moved_text();Brandt matrices and theta series over global function fields

Chih-Yun Chuang, Ting-Fang Lee, Fu-Tsun Wei and Jing Yu

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1117
ISBNs: 978-1-4704-1419-1 (print); 978-1-4704-2501-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1117
Published electronically: January 5, 2015
Keywords:Function fields, Brandt matrices, automorphic forms on $\GL _2$, theta series

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Chapters

• Chapter 1. Introduction
• Chapter 2. Brandt matrices and definite Shimura curves
• Chapter 3. The basis problem for Drinfeld type automorphic forms
• Chapter 4. Metaplectic forms and Shintani-type correspondence
• Chapter 5. Trace formula of Brandt matrices

### Abstract

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 $\infty$, we 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 our theta series, and thereby contributes to the so-called basis problem. Restricting the norm forms to pure quaternions, we obtain another family of theta series which are automorphic functions on the metaplectic group, and results in a Shintani-type correspondence between Drinfeld type forms and metaplectic forms.