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Stability results for local zeta functions of groups algebras, and modules

  • TOBIAS ROSSMANN (a1) (a2)
Abstract
Abstract

Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

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[1] Avni N., Klopsch B., Onn U. and Voll C. Representation zeta functions of compact p-adic analytic groups and arithmetic groups. Duke Math. J. 162 (2013), no. 1, 111197.
[2] Denef J. Report on Igusa's local zeta function. Astérisque 201–203 (1991), Exp. No. 741, 359–386 (1992). Séminaire Bourbaki, Vol. 1990/91.
[3] Denef J. and Loeser F. Caractéristiques d'Euler-Poincaré, fonctions zêta locales et modifications analytiques. J. Amer. Math. Soc. 5 (1992), no. 4, 705720.
[4] Denef J. and Loeser F. Motivic Igusa zeta functions. J. Algebraic Geom. 7 (1998), no. 3, 505537.
[5] Denef J. and Meuser D. A functional equation of Igusa's local zeta function. Amer. J. Math. 113 (1991), no. 6, 11351152.
[6] du Sautoy M. P. F. and Grunewald F. J. Analytic properties of zeta functions and subgroup growth. Ann. of Math. (2) 152 (2000), no. 3, 793833.
[7] du Sautoy M. P. F. and Loeser F. Motivic zeta functions of infinite-dimensional Lie algebras. Selecta Math. (N.S.) 10 (2004), no. 2, 253303.
[8] du Sautoy M. P. F. and Woodward L. Zeta functions of groups and rings. Lecture Notes in Math. vol. 1925 (Springer-Verlag, Berlin, 2008).
[9] Grunewald F. J., Segal D. and Smith G. C. Subgroups of finite index in nilpotent groups. Invent. Math. 93 (1988), no. 1, 185223.
[10] Katz N. M. Review of ℓ-adic cohomology. Motives (Seattle, WA, 1991), (1994), pp. 2130.
[11] Klopsch B. Representation growth and representation zeta functions of groups. Note Mat. 33 (2013), no. 1, 107120.
[12] Nunley C. and Magid A. Simple representations of the integral Heisenberg group. Classical groups and related topics (Beijing, 1987), (1989), pp. 8996.
[13] Rossmann T. Computing topological zeta functions of groups, algebras, and modules, I. Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 10991134.
[14] Rossmann T. Computing topological zeta functions of groups, algebras, and modules, II. J. Algebra 444 (2015), 567605.
[15] Rossmann T. Topological representation zeta functions of unipotent groups. J. Algebra 448 (2016), 210237.
[16] Segal D. Ideals of finite index in a polynomial ring. Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 189, 8392.
[17] Serre J.-P. Lectures on NX (p). Chapman & Hall/CRC Res. Not. Math. vol. 11 (CRC Press, Boca Raton, FL, 2012).
[18] Solomon L. Zeta functions and integral representation theory. Adv. Math. 26 (1977), no. 3, 306326.
[19] Stasinski A. and Voll C. Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B . Amer. J. Math. 136 (2014), no. 2, 501550.
[20] Voll C. Functional equations for zeta functions of groups and rings. Ann. of Math. (2) 172 (2010), no. 2, 11811218.
[21] Voll C. Zeta functions of groups and rings–-recent developments. Groups St Andrews (2013), (2015), pp. 469492.
[22] Woodward L. Zeta functions of groups: computer calculations and functional equations. D.Phil. thesis, University of Oxford, 2005.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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