Arithmetic and analytic theories of quadratic forms and Clifford groups / Goro Shimura.

Format:

Book

Author/Creator:

Shimura, Gorō, 1930-2019, author.

Series:

Mathematical surveys and monographs ; volume 109.

Mathematical surveys and monographs ; volume 109

Language:

English

Subjects (All):

Forms, Quadratic.

Linear algebraic groups.

Number theory.

Physical Description:

1 online resource (290 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2004]

Language Note:

English

Summary:

In this book, award-winning author Goro Shimura treats new areas and presents relevant expository material in a clear and readable style. Topics include Witt's theorem and the Hasse principle on quadratic forms, algebraic theory of Clifford algebras, spin groups, and spin representations. He also includes some basic results not readily found elsewhere. The two principle themes are: (1) Quadratic Diophantine equations; (2) Euler products and Eisenstein series on orthogonal groups and Clifford groups. The starting point of the first theme is the result of Gauss that the number of primitive representations of an integer as the sum of three squares is essentially the class number of primitive binary quadratic forms. Presented are a generalization of this fact for arbitrary quadratic forms over algebraic number fields and various applications. For the second theme, the author proves the existence of the meromorphic continuation of a Euler product associated with a Hecke eigenform on a Clifford or an orthogonal group. The same is done for an Eisenstein series on such a group. Beyond familiarity with algebraic number theory, the book is mostly self-contained. Several standard facts are st

Contents:

""Table of Contents""; ""Preface""; ""Notation and Terminology""; ""Introduction""; ""Chapter I. Algebraic theory of quadratic forms, Clifford algebras, and spin groups""; ""1. Quadratic forms and associative algebras""; ""2. Clifford algebras""; ""3. Clifford groups and spin groups""; ""4. Parabolic subgroups""; ""Chapter II. Quadratic forms, Clifford algebras, and spin groups over a local or global field""; ""5. Orders and ideals in an algebra""; ""6. Quadratic forms over a local field""; ""7. Lower-dimensional cases and the Hasse principle""

""8. Part I. Clifford groups over a local field""""8. Part II. Formal Hecke algebras and formal Euler factors""; ""9. Orthogonal, Clifford, and spin groups over a global field""; ""Chapter III. Quadratic Diophantine equations""; ""10. Quadratic Diophantine equations over a local field""; ""11. Quadratic Diophantine equations over a global field""; ""12. The class number of an orthogonal group and sums of squares""; ""13. Nonscalar quadratic Diophantine equations; Connection with the mass formula; A historical perspective""; ""Chapter IV. Groups and symmetric spaces over R""

""14. Clifford and spin groups over R The case of signature (1,m)""; ""15. The case of signature (2,m)""; ""16. Orthogonal groups over R and symmetric spaces""; ""Chapter V. Euler products and Eisenstein series on orthogonal groups""; ""17. Automorphic forms and Euler products on an orthogonal group""; ""18. Eisenstein series on o[sup(w)]""; ""19. Eisenstein series on ...""; ""20. Arithmetic description of the pullback of an Eisenstein series""; ""21. Analytic continuation of Euler products and Eisenstein series""; ""Chapter VI. Euler products and Eisenstein series on Clifford groups""

""22. Euler products on G[sup(+)](V)""""23. Eisenstein series on . . .""; ""24. Eisenstein series of general types on a Clifford group""; ""25. Euler products for holomorphic forms on a Clifford group""; ""26. Proof of the last main theorem""; ""Appendix""; ""A1. Differential operators on a semisimple Lie group""; ""A2. Eigenvalues of integral operators""; ""A3. Structure of Clifford algebras over R""; ""A4. An embedding of G[sup(1)](v) into a symplectic group""; ""A5. Spin representations and Lie algebras""; ""References""; ""Frequently used symbols""; ""Index""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 272-273) and index.

Description based on print version record.

ISBN:

1-4704-1336-1