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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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##### References:
 [1] \beginbbook \bauthor\binitsA. \bsnmBorel, \bbtitleLinear algebraic groups, \bedition2nd ed., \bsertitleGraduate Texts in Mathematics, vol. \bseriesno126, \bpublisherSpringer, \blocationNew York, \byear1991. \endbbook \OrigBibText Borel, Armand, Linear algebraic groups, Graduate Texts in Mathematics, 126, 2, Springer-Verlag, New York, 1991. \endOrigBibText \bptokstructpyb \endbibitem [2] \beginbbook \bauthor\binitsA. \bsnmBorel, \bbtitleIntroduction aux groupes arithmétiques, \bsertitlePublications de l’Institut de Mathématique de l’Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, vol. \bseriesno1341, \bpublisherHermann, \blocationParis, \byear1969. \endbbook \OrigBibText Borel, Armand, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969, 125. \endOrigBibText \bptokstructpyb \endbibitem [3] \beginbbook \bauthor\binitsA. \bsnmBorel and \bauthor\binitsN. \bsnmWallach, \bbtitleContinuous cohomology, discrete subgroups, and representations of reductive groups, \bedition2nd ed., \bsertitleMathematical Surveys and Monographs, vol. \bseriesno67, \bpublisherAmerican Mathematical Society, \blocationProvidence, RI, \byear2000. \endbbook \OrigBibText Borel, A., Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical Surveys and Monographs, 67, 2, American Mathematical Society, Providence, RI, 2000. \endOrigBibText \bptokstructpyb \endbibitem [4] \beginbchapter \bauthor\binitsP. \bsnmCartier, \bctitleRepresentations of $$p$$-adic groups: A survey, \bbtitleAutomorphic forms, representations and $$L$$-functions \bmisc(Proc. Sympos. Pure Math., Oregon State Univ. \bconflocationCorvallis, OR, \bconfdate1977), \bsertitleProc. Sympos. Pure Math., vol. \bseriesnoXXXIII, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1979, pp. page 111-\blpage155. \endbchapter \OrigBibText Cartier, P., Representations of $$p$$-adic groups: a survey, Automorphic forms, representations and $$L$$-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., (1977), Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, 111-155. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0421.22010 [5] \beginbarticle \bauthor\binitsA. \bsnmDeitmar, \batitleGeometric zeta functions of locally symmetric spaces, \bjtitleAmer. J. Math. \bvolume122 (\byear2000), no. \bissue5, page 887-\blpage926. \endbarticle \OrigBibText Deitmar, Anton, Geometric zeta functions of locally symmetric spaces, Amer. J. Math., 122, (2000), 5, 887-926. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0369.46061 [6] \beginbarticle \bauthor\binitsA. \bsnmDeitmar, \batitleA prime geodesic theorem for higher rank spaces, \bjtitleGeom. Funct. Anal. \bvolume14 (\byear2004), no. \bissue6, page 1238-\blpage1266. \endbarticle \OrigBibText Deitmar, A., A prime geodesic theorem for higher rank spaces, Geom. Funct. Anal., 14, (2004), 6, 1238-1266. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1102.11028 [7] \beginbarticle \bauthor\binitsA. \bsnmDeitmar, \batitleLefschetz formulae for $$p$$-adic groups, \bjtitleChin. Ann. Math. Ser. B \bvolume28 (\byear2007), no. \bissue4, page 463-\blpage474. \endbarticle \OrigBibText Deitmar, Anton, Lefschetz formulae for $$p$$-adic groups, Chin. Ann. Math. Ser. B, 28, (2007), 4, 463-474. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1223.11143 [8] \beginbarticle \bauthor\binitsY. \bsnmIhara, \batitleOn discrete subgroups of the two by two projective linear group over $$\mathfrak{p}$$-adic fields, \bjtitleJ. Math. Soc. Japan \bvolume18 (\byear1966), page 219-\blpage235. \endbarticle \OrigBibText Ihara, Yasutaka, On discrete subgroups of the two by two projective linear group over $$\mathfrak{p}$$-adic fields, J. Math. Soc. Japan, 18, (1966), 219-235. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0158.27702 [9] \beginbarticle \bauthor\binitsM.-H. \bsnmKang and \bauthor\binitsW.-C. W. \bsnmLi, \batitleThe zeta functions of complexes from $$\mathrm{PGL}(3)$$, \bjtitleAdv. Math. \bvolume256 (\byear2014), page 46-\blpage103. \endbarticle \OrigBibText Kang, Ming-Hsuan, Li, Wen-Ching Winnie, The zeta functions of complexes from $${\mathrm PGL}(3)$$, Advances in Mathematics, 256, 46-103, (2014) \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1328.22008 [10] \beginbarticle \bauthor\binitsM.-H. \bsnmKang, \bauthor\binitsW.-C. W. \bsnmLi and \bauthor\binitsC.-J. \bsnmWang, \batitleThe zeta functions of complexes from $$\mathrm{PGL}(3)$$: A representation-theoretic approach, \bjtitleIsrael J. Math. \bvolume177 (\byear2010), page 335-\blpage348. \endbarticle \OrigBibText Kang, Ming-Hsuan, Li, Wen-Ching Winnie, Wang, Chian-Jen, The zeta functions of complexes from $${\mathrm PGL}(3)$$: a representation-theoretic approach, Israel J. Math., 177, (2010), 335-348. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1230.05286 [11] \beginbarticle \bauthor\binitsR. E. \bsnmKottwitz, \batitleTamagawa numbers, \bjtitleAnn. of Math. (2) \bvolume127 (\byear1988), no. \bissue3, page 629-\blpage646. \endbarticle \OrigBibText Kottwitz, Robert E., Tamagawa numbers, Ann. of Math. (2), 127, (1988), 3, 629-646. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0678.22012 [12] \beginbarticle \bauthor\binitsW.-C. W. \bsnmLi, \batitleRamanujan hypergraphs, \bjtitleGeom. Funct. Anal. \bvolume14 (\byear2004), no. \bissue2, page 380-\blpage399. \endbarticle \OrigBibText Li, Wen-Ching Winnie, Ramanujan hypergraphs, Geom. Funct. Anal., 14, (2004), 2, 380-399. \endOrigBibText \bptokstructpyb \endbibitem [13] \beginbarticle \bauthor\binitsA. \bsnmLubotzky, \bauthor\binitsB. \bsnmSamuels and \bauthor\binitsU. \bsnmVishne, \batitleRamanujan complexes of type $$A_d$$, probability in mathematics, \bjtitleIsrael J. Math. \bvolume149 (\byear2005), page 267-\blpage299. \endbarticle \OrigBibText Lubotzky, Alexander, Samuels, Beth, Vishne, Uzi, Ramanujan complexes of type $$A_d$$, Probability in mathematics, Israel J. Math., 149, (2005), 267-299. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1087.05036 [14] \beginbchapter \bauthor\binitsT. \bsnmSunada, \bctitle$$L$$-Functions in geometry and some applications, \bbtitleCurvature and topology of Riemannian manifolds (\bconflocationKatata, \bconfdate1985), \bsertitleLecture Notes in Math., vol. \bseriesno1201, \bpublisherSpringer, \blocationBerlin, \byear1986, pp. page 266-\blpage284. \endbchapter \OrigBibText Sunada, Toshikazu, $$L$$-functions in geometry and some applications, Curvature and topology of Riemannian manifolds, Katata, 1985, Lecture Notes in Math., 1201, Springer, Berlin, 1986, 266-284. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0605.58046 [15] \beginbchapter \bauthor\binitsJ. \bsnmTits, \bctitleReductive groups over local fields, \bbtitleAutomorphic forms, representations and $$L$$-functions \bmisc(Proc. Sympos. Pure Math., Oregon State Univ. \bconflocationCorvallis, OR, \bconfdate1977), \bsertitleProc. Sympos. Pure Math., vol. \bseriesnoXXXIII, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1979, pp. page 29-\blpage69. \endbchapter \OrigBibText Tits, J., Reductive groups over local fields, Automorphic forms, representations and $$L$$-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, 29-69. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0415.20035 [16] \beginbarticle \bauthor\binitsJ. A. \bsnmWolf, \batitleDiscrete groups, symmetric spaces, and global holonomy, \bjtitleAmer. J. Math. \bvolume84 (\byear1962), page 527-\blpage542. \endbarticle \OrigBibText Wolf, Joseph A., Discrete groups, symmetric spaces, and global holonomy, Amer. J. Math., 84, (1962), 527-542. \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0116.38602
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