부산대학교 도서관

The Selberg zeta function is defined by counting closed geodesics in a Riemann surface. Similarly, the Ihara zeta function is obtained by counting closed geodesics in a graph. Zeta functions of this kind, coming from geometric data, are referred to as geometric zeta functions. For a finite graph, the Ihara zeta function is the same as the Hasse-Weil zeta function of the associated Shimura curve [Y. Ihara, J. Math. Soc. Japan {\bf 18} (1966), 219--235; MR0223463], providing a link to the arithmetically defined zeta functions. This fact follows from the so-called Ihara formula. For the case of $PGL_3$ this was generalized in [M.-H. Kang and W.-C.~W. Li, Adv. Math. {\bf 256} (2014), 46--103; MR3177290] and [M.-H. Kang, W.-C.~W. Li and C.-J. Wang, Israel J. Math. {\bf 177} (2010), 335--348; MR2684424]. The goal of this paper is to generalize Ihara's approach to a higher-dimensional case. \par The idea is to take the trace formula approach. More precisely, using the Lefschetz formula proved in [A. Deitmar, Chin. Ann. Math. Ser. B {\bf 28} (2007), no.~4, 463--474; MR2348458], a several-variable zeta function is defined. It is associated to a discrete cocompact subgroup $\Gamma$ of a semi-simple linear algebraic group $G$ defined over a non-archimedean local field and a certain finite-dimensional representation. The analytic continuation and rationality of this zeta function is proved. It is a priori defined in terms of conjugacy classes in $\Gamma$, but it turns out that these actually count the closed geodesics in $\Gamma\backslash\Cal{B}$, where $\Cal{B}$ is the Bruhat-Tits building of $G$. This provides the relation to geometric data and geometric zeta functions in higher rank. \par Finally, in the case of $G=PGL_3$, the geometric zeta functions obtained in the paper are compared to those of [M.-H. Kang, W.-C. W. Li, C.-J. Wang, op. cit.].