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Voll, Christopher

Functional equations for zeta functions of groups and rings. (English) Zbl 1314.11057

Ann. Math. (2) 172, No. 2, 1181-1218 (2010).
Summary: We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or \(\mathcal T\)-)groups, and the normal zeta functions of \(\mathcal T\)-groups of class 2. We deduce our theorems from a “blueprint result” on certain \(p\)-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to “linearise” the problems of counting subgroups and representations in \(\mathcal T\)-groups, respectively.

Cited in 32 Documents

MSC:

11M41 Other Dirichlet series and zeta functions
11S40 Zeta functions and \(L\)-functions
20E07 Subgroup theorems; subgroup growth

Keywords:

subgroup growth; representation growth; nilpotent groups; Igusa’s local zeta function; \(p\)-adic integration; local functional equations; Kirillov theory
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\textit{C. Voll}, Ann. Math. (2) 172, No. 2, 1181--1218 (2010; Zbl 1314.11057)
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References:

[1] M. N. Berman, ”Uniformity and functional equations for local zeta functions of \(\mathfrakK\)-split algebraic groups,” , preprint , 2008.
[2] C. W. Curtis and I. Reiner, Methods of Representation Theory, With Applications to Finite Groups and Orders, New York: John Wiley & Sons, 1981, vol. I. · Zbl 0469.20001
[3] J. Denef, ”On the degree of Igusa’s local zeta function,” Amer. J. Math., vol. 109, iss. 6, pp. 991-1008, 1987. · Zbl 0659.14017
[4] J. Denef, ”Report on Igusa’s local zeta function,” in Séminaire Bourbaki, Vol. \(1990/91\), , 1992, vol. 201-203, pp. 359-386. · Zbl 0749.11054
[5] J. Denef and D. Meuser, ”A functional equation of Igusa’s local zeta function,” Amer. J. Math., vol. 113, iss. 6, pp. 1135-1152, 1991. · Zbl 0749.11053
[6] M. du Sautoy, ”A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups,” Israel J. Math., vol. 126, pp. 269-288, 2001. · Zbl 0998.20033
[7] M. du Sautoy, ”Counting subgroups in nilpotent groups and points on elliptic curves,” J. Reine Angew. Math., vol. 549, pp. 1-21, 2002. · Zbl 1001.20032
[8] M. du Sautoy and F. Grunewald, ”Analytic properties of zeta functions and subgroup growth,” Ann. of Math., vol. 152, iss. 3, pp. 793-833, 2000. · Zbl 1006.11051
[9] M. du Sautoy and F. Grunewald, ”Zeta functions of groups and rings,” in Proc. Internat. Congress of Mathematicians, Eur. Math. Soc., Zürich, 2006, vol. II, pp. 131-149. · Zbl 1112.20024
[10] M. du Sautoy and A. Lubotzky, ”Functional equations and uniformity for local zeta functions of nilpotent groups,” Amer. J. Math., vol. 118, iss. 1, pp. 39-90, 1996. · Zbl 0864.20020
[11] M. du Sautoy and D. Segal, ”Zeta functions of groups,” in New Horizons in Pro-\(p\) Groups, Boston, MA: Birkhäuser, 2000, vol. 184, pp. 249-286. · Zbl 1188.11047
[12] M. du Sautoy and G. Taylor, ”The zeta function of \(\mathfrak{sl}_2\) and resolution of singularities,” Math. Proc. Cambridge Philos. Soc., vol. 132, iss. 1, pp. 57-73, 2002. · Zbl 1020.11074
[13] M. du Sautoy and L. Woodward, Zeta Functions of Groups and Rings, New York: Springer-Verlag, 2008, vol. 1925. · Zbl 1151.11005
[14] F. J. Grunewald, D. Segal, and G. C. Smith, ”Subgroups of finite index in nilpotent groups,” Invent. Math., vol. 93, iss. 1, pp. 185-223, 1988. · Zbl 0651.20040
[15] R. E. Howe, ”Kirillov theory for compact \(p\)-adic groups,” Pacific J. Math., vol. 73, iss. 2, pp. 365-381, 1977. · Zbl 0385.22007
[16] R. E. Howe, ”On representations of discrete, finitely generated, torsion-free, nilpotent groups,” Pacific J. Math., vol. 73, iss. 2, pp. 281-305, 1977. · Zbl 0387.22005
[17] E. Hrushovski and B. Martin, Zeta functions from definable equivalence relations, 2004.
[18] J. Igusa, ”Universal \(p\)-adic zeta functions and their functional equations,” Amer. J. Math., vol. 111, iss. 5, pp. 671-716, 1989. · Zbl 0707.14016
[19] A. Jaikin-Zapirain, ”Zeta function of representations of compact \(p\)-adic analytic groups,” J. Amer. Math. Soc., vol. 19, iss. 1, pp. 91-118, 2006. · Zbl 1092.20023
[20] E. I. Khukhro, \(p\)-Automorphisms of Finite \(p\)-Groups, Cambridge: Cambridge Univ. Press, 1998, vol. 246. · Zbl 0897.20018
[21] B. Klopsch, ”Zeta functions related to the pro-\(p\) group \({ SL}_1(\Delta_p)\),” Math. Proc. Cambridge Philos. Soc., vol. 135, iss. 1, pp. 45-57, 2003. · Zbl 1076.20018
[22] B. Klopsch and C. Voll, ”Zeta functions of three-dimensional \(p\)-adic Lie algebras,” Math. Z., vol. 263, pp. 195-210, 2009. · Zbl 1193.11083
[23] A. Lubotzky and A. R. Magid, ”Varieties of representations of finitely generated groups,” Mem. Amer. Math. Soc., vol. 58, iss. 336, p. xi, 1985. · Zbl 0598.14042
[24] A. Lubotzky and D. Segal, Subgroup Growth, Basel: Birkhäuser, 2003, vol. 212. · Zbl 1071.20033
[25] C. Nunley and A. Magid, ”Simple representations of the integral Heisenberg group,” in Classical Groups and Related Topics, Providence, RI: Amer. Math. Soc., 1989, vol. 82, pp. 89-96. · Zbl 0698.22009
[26] D. Segal, Polycyclic Groups, Cambridge: Cambridge Univ. Press, 1983, vol. 82. · Zbl 0516.20001
[27] J. Serre, Lie Algebras and Lie Groups, Second ed., New York: Springer-Verlag, 1992, vol. 1500. · Zbl 0742.17008
[28] R. P. Stanley, Enumerative Combinatorics. Vol. \(1\), Cambridge: Cambridge Univ. Press, 1997, vol. 49. · Zbl 0889.05001
[29] W. Veys and W. A. Zúñiga-Galindo, ”Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra,” Trans. Amer. Math. Soc., vol. 360, iss. 4, pp. 2205-2227, 2008. · Zbl 1222.11141
[30] C. Voll, ”Zeta functions of groups and enumeration in Bruhat-Tits buildings,” Amer. J. Math., vol. 126, iss. 5, pp. 1005-1032, 2004. · Zbl 1076.11050
[31] C. Voll, ”Functional equations for local normal zeta functions of nilpotent groups,” Geom. Funct. Anal., vol. 15, iss. 1, pp. 274-295, 2005. · Zbl 1135.11331
[32] C. Voll, ”Counting subgroups in a family of nilpotent semi-direct products,” Bull. London Math. Soc., vol. 38, iss. 5, pp. 743-752, 2006. · Zbl 1120.11040
[33] J. Włodarczyk, ”Simple Hironaka resolution in characteristic zero,” J. Amer. Math. Soc., vol. 18, iss. 4, pp. 779-822, 2005. · Zbl 1084.14018
[34] L. Woodward, ”Zeta functions of groups: computer calculations and functional equations,” PhD Thesis , University of Oxford, 2005.
[35] R. I. Liu, ”Counting subrings of \(\mathbbZ^n\) of index \(k\),” J. Combinatorial Theory, vol. 114, pp. 278-299, 2007. · Zbl 1117.11006
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