Selberg-Ihara's Zeta function for p-adic Discrete Groups

Abstract

This chapter focuses on Selberg–Ihara's zeta function for p-adic discrete groups. In [SeI], a zeta function ZΓ(s) has been introduced and proved to have many important properties that resemble those of usual L-functions, such as Euler product, functional equation, and analogue of Riemann hypothesis. This function, called with the name of Selberg, is generalized to any discrete subgroup Γ of a semi-simple Lie group of R-rank one. An analogue of ZΓ(s) was introduced by Ihara, for a cocompact torsion-free discrete subgroup Γ of PSL(2,K) or PL(2,K), where K is a p-adic field. The chapter presents an extension of Ihara's results to the case when G is a semi-simple algebraic group over a p-adic field K and Γ is a discrete subgroup of G. The chapter discusses p-adic algebraic groups and the structure of the discrete subgroups Γ.