Ebook: Introduction to Siegel modular forms and Dirichlet series

Introduction to Siegel Modular Forms and Dirichlet Series gives a concise and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms in one variable. It serves to attract young researchers to this beautiful field and makes the initial steps more pleasant. It treats a number of questions that are rarely mentioned in other books. It is the first and only book so far on Siegel modular forms that introduces such important topics as analytic continuation and the functional equation of spinor zeta functions of Siegel modular forms of genus two.

Unique features include:

* New, simplified approaches and a fresh outlook on classical problems

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

* The action of Hecke operators on Siegel modular forms

* Applications of Hecke operators to a study of the multiplicative properties of Fourier coefficients of modular forms

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

*Numerous exercises

Anatoli Andrianov is a leading researcher at the St. Petersburg branch of the Steklov Mathematical Institute of the Russian Academy of Sciences. He is well known for his works on the arithmetic theory of automorphic functions and quadratic forms, a topic on which he has lectured at many universities around the world.

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).