Quadratic forms and Hecke operators

More about the book

The classical theory of quadratic forms has revealed remarkable multiplicative properties regarding the integral representations of integers by positive definite integral quadratic forms, explained by Hecke operators. However, no such properties were known for the integral representations of quadratic forms in multiple variables. Despite this, the appeal of Hecke operators led to efforts to explore their application in the realm of modular forms with several variables. This exploration has yielded fruitful results, uncovering several multiplicative properties of integral representations of quadratic forms by quadratic forms. The theory has now matured, warranting a concise and up-to-date report to lay a solid foundation for future advancements. This book aims to serve as a self-contained textbook detailing the contemporary state of Hecke operators on spaces of holomorphic modular forms of integral weight (specifically, Siegel modular forms) for congruence subgroups of integral symplectic groups. Additionally, it can be utilized for introductory studies of modular forms in one or several variables, as well as theta-series of positive definite integral quadratic forms.