This volume aims to present a straightforward and easily accessible survey of the analytic theory of quadratic forms. Written at an elementary level, the book provides a sound basis from which the reader can study advanced works and undertake original research. Roughly half a century ago C.L. Siegel discovered a new type of automorphic forms in several variables in connection with his famous work on the analytic theory of quadratic forms. Since then Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the recent arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The text is based on the author's lectures given over a number of years and is intended for a one semester graduate course, although it can serve equally well for self study . The only prerequisites are a knowledge of algebra, number theory and complex analysis.
Author(s): Helmut Klingen
Series: Cambridge Studies in Advanced Mathematics 20
Publisher: CUP
Year: 1990
Language: English
Pages: 172
Contents......Page 7
Preface......Page 9
1 The symplectic group......Page 11
2 Minkowski's reduction theory......Page 20
3 Fundamental sets of Siegel's modular group......Page 37
4 The linear space of modular forms......Page 53
5 Eisenstein series and the Siegel operator......Page 64
6 Cusp forms and Poincare series......Page 85
7 Non-cusp forms......Page 103
8 Singular modular forms and theta-series......Page 109
9 The graded ring of modular forms of degree two......Page 122
10 Quotients of modular forms......Page 134
11 Pseudoconcavity......Page 139
VI Dirichlet series......Page 153
12 Dirichlet series associated with modular forms and the Mellin-transform......Page 154
13 Analytic continuation and the functional equation......Page 158
Bibliography......Page 167
Index......Page 171
