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 Z Γ ( 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 Z Γ ( 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 .

Locations

• Project Euclid (Cornell University) - View - PDF
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