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Geometric zeta functions for higher rank \(p\)-adic groups. (English) Zbl 1377.11102
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 18, 219–235 (1966; Zbl 0158.27702)], providing a link to the arithmetically defined zeta functions. This fact follows from the so-called Ihara formula. For the case of \(\mathrm{PGL}_3\) this was generalized in [the second author and W.-C. W. Li, Adv. Math. 256, 46–103 (2014; Zbl 1328.22008)] and [the second author et al., Isr. J. Math. 177, 335–348 (2010; Zbl 1230.05286)]. The goal of this paper is to generalize Ihara’s approach to a higher-dimensional case. The idea is to take the trace formula approach. More precisely, using the Lefschetz formula proved in [the first author, Chin. Ann. Math., Ser. B 28, No. 4, 463–474 (2007; Zbl 1223.11143)], 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 \(\Gamma\backslash\mathcal{B}\), where \(\mathcal{B}\) is the Bruhat-Tits building of \(G\). This provides the relation to geometric data and geometric zeta functions in higher rank.
Finally, in the case of \(G=\mathrm{PGL}_3\), the geometric zeta functions obtained in the paper are compared to those of [the second author, Zbl 1230.05286].

MSC:
11M41 Other Dirichlet series and zeta functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
20E42 Groups with a \(BN\)-pair; buildings
22E50 Representations of Lie and linear algebraic groups over local fields
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