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Brandt Matrices and Theta Series over Global Function Fields.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2015Copyright date: ©2015Edition: 1st edDescription: 1 online resource (76 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470425012
Subject(s): Genre/Form: Additional physical formats: Print version:: Brandt Matrices and Theta Series over Global Function FieldsDDC classification:
  • 512.70000000000005
LOC classification:
  • QA196.B736 2015
Online resources:
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.
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.
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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.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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