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Brandt matrices and theta series over global function fields
About this Title
Chih-Yun Chuang, Department of Mathematics, National Tsing-Hua University, No. 101, Sec. 2, KuangFu Rd., Hsinchu 30013 Taiwan, Ting-Fang Lee, Department of Mathematics, National Tsing-Hua University, No. 101, Sec. 2, KuangFu Rd., Hsinchu 30013 Taiwan, Fu-Tsun Wei, Department of Mathematics, National Tsing-Hua University, No. 101, Sec. 2, KuangFu Rd., Hsinchu 30013 Taiwan and Jing Yu, Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan
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: https://doi.org/10.1090/memo/1117
Published electronically: January 5, 2015
Keywords: Function fields,
Brandt matrices,
automorphic forms on $\operatorname {GL}_2$,
theta series,
metaplectic forms
MSC: Primary 11R58, 11F27, 11F30, 11F37, 11F41
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 Shintani-type correspondence
- 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.- Marleen Denert and Jan Van Geel, The class number of hereditary orders in non-Eichler algebras over global function fields, Math. Ann. 282 (1988), no. 3, 379–393. MR 967020, DOI 10.1007/BF01460041
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