Introduction to Siegel Modular Forms and Dirichlet Series

This is intended for a graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The author’s aim is to present a concise and self-contained introduction to an important and developing area of number theory that will serve to attract young researchers to this beautiful field.

Topics include:

* analytical properties of radial Dirichlet series attached to modular forms of genuses 1 and 2;

* the abstract theory of Hecke–Shimura rings for symplectic and related groups;

* action of Hecke operators on Siegel modular forms;

* applications of Hecke operators to a study of multiplicative properties of Fourier coefficients of modular forms;

* Hecke zeta functions of modular forms in one variable and to spinor (or Andrianov) zeta functions of Siegel modular forms of genus two;

* the proof of analytical continuation and functional equation (under certain assumptions) for Euler products associated with modular forms of genus two.

This text contains a number of exercises and the only prerequisites are standard courses in Algebra and Calculus (one and several variables).