EULER PRODUCTS CORRESPONDING TO SIEGEL MODULAR FORMS OF GENUS 2

Abstract

In this article we construct a theory of Dirichlet series with Euler product expansions corresponding to analytic automorphic forms for the integral symplectic group in genus 2; in Chapter 2 we establish a connection between the eigenvalues of the Hecke operators on the spaces of such forms with the Fourier coefficients of the eigenfunctions (Theorem 2.4.1); in Chapter 3 we demonstrate the possibility of analytic continuation to the entire complex plane and derive a functional equation for Euler products corresponding to the eigenfunctions of the Hecke operators (Theorem 3.1.1). Chapter 1 contains a survey of the present state of the theory of Euler products for Siegel modular forms of arbitrary genus n, including a sketch of the classical Hecke theory for the case n = 1.