Zeta-functions of simple algebras with non-abelian characters

We prove that the zeta-functions can be continued analytically to the whole plane and obtain a functional equation for the zeta-functions with non-abelian characters of arbitrary simple central algebras over fields of algebraic numbers. By “non-abelian characters” we mean positive-definite zonal spherical functions on the group of ideles of the algebra in question that belong to automorphic functions on the group of ideles. The proofs are based on the Poisson formula for the additive group of ideles of the algebra, adapted for a specially constructed weight function (the “excision” weight function), which vanishes together with its Fourier transform on elements of norm zero (theta-formula). The paper consists of four chapters. Chapter 1 contains a survey of the necessary facts on simple algebras over local fields and their zeta-functions. Chapter 2 is fundamental and deals with the construction of the “excision” weight functions. Chapter 3 contains a survey of the basic facts on restricted direct products of locally compact groups. In Chapter 4 the technique of “excision” weight functions and Fourier analysis on the group of ideles is applied to the study of the zeta-functions of simple algebras over number fields.