RATIONALITY THEOREMS FOR HECKE SERIES AND ZETA FUNCTIONS OF THE GROUPS<i>GL_n</i>and<i>SP_n</i>OVER LOCAL FIELDS

Abstract

We prove the rationality of multiple Hecke series and multiple zeta functions of the multiplicative groups of simple algebras over local fields, as well as the Shimura conjecture concerning the rationality of Hecke series and zeta functions of symplectic groups over local fields.