Lectures On Siegel Modular Forms And Representation By Quadratic Forms

Lectures on Siegel Modular Forms and Representation by Quadratic Forms

Lectures on Siegel Modular Forms and Representation by Quadratic Forms

These are based on my lectures at the Tata Institute of Fundamental Research in 1983-84. They are concerned with the problern of representation of positive definite quadratic forms by other such forms. § 1.6 and Chapter 2 are added, besides lectures at the Institute, by Professor Raghavan (who also wrote up §§ 1.1 - 1.4) and myself respectively. I would like to thank Professor Raghavan and the Tata Institute for their hospitality. Y. KITAOKA CHAPTER 1. FOURIER COEFFICIENTS OF SIEGEL MODULAR FO~\IS Introduction 1 § 1.1. Estimates for Fourier coefficients of cusp forms of degree 1 10 § 1.2. Reduction theory 25 § 1. 3. Minkowski reduced domain 35 § 1.4. Estimation of Fourier coefficients of modular forms of degree n 45 § 1. 5. Generalization of Klocsterman's method to the case of degree 2 83 § 1.6. Estimation of Fourier coefficients of modular forms 117 § 1. 7. Primitive representations 161 CHAPTER 2. ARITHMETIC OF QUADRATIC FORMS § Notation and terminology 2.0 167 § 2.1 Quadratic modules over Q 169 p § 2.1.2. Modular and maximal lattices 170 § 2.1.3. Jordan splittings 175 § 178 2.1.4. Extension theorems § 185 2.2. The spinor norm § 2.3. Hasse-Minkowski theorem 191 § 2.4. Integta1 theory of quadratic forms 192 225 REFERENCES CHAFTER 1 FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS INTRODUCTION.

Siegel Modular Forms

This monograph introduces two approaches to studying Siegel modular forms: the classical approach as holomorphic functions on the Siegel upper half space, and the approach via representation theory on the symplectic group. By illustrating the interconnections shared by the two, this book fills an important gap in the existing literature on modular forms. It begins by establishing the basics of the classical theory of Siegel modular forms, and then details more advanced topics. After this, much of the basic local representation theory is presented. Exercises are featured heavily throughout the volume, the solutions of which are helpfully provided in an appendix. Other topics considered include Hecke theory, Fourier coefficients, cuspidal automorphic representations, Bessel models, and integral representation. Graduate students and young researchers will find this volume particularly useful. It will also appeal to researchers in the areaas a reference volume. Some knowledge of GL(2) theory is recommended, but there are a number of appendices included if the reader is not already familiar.