Computing local zeta functions of groups, algebras, and modules
HTML articles powered by AMS MathViewer
- by Tobias Rossmann PDF
- Trans. Amer. Math. Soc. 370 (2018), 4841-4879 Request permission
Abstract:
We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with $\mathfrak {gl}_2(\mathbf {Q})$. Finally, we introduce and compute examples of graded subobject zeta functions.References
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608, DOI 10.1090/gsm/003
- Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, Representation zeta functions of compact $p$-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), no. 1, 111–197. MR 3011874, DOI 10.1215/00127094-1959198
- V. Baldoni, N. Berline, J. A. De Loera, B. Dutra, M. Köppe, S. Moreinis, G. Pinto, M. Vergne, and J. Wu, A user’s guide for LattE integrale v1.7.3, 2015. Software package. LattE is available at http://www.math.ucdavis.edu/~latte/.
- Alexander Barvinok, Integer points in polyhedra, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2455889, DOI 10.4171/052
- Alexander Barvinok and James E. Pommersheim, An algorithmic theory of lattice points in polyhedra, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97) Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 91–147. MR 1731815
- Alexander Barvinok and Kevin Woods, Short rational generating functions for lattice point problems, J. Amer. Math. Soc. 16 (2003), no. 4, 957–979. MR 1992831, DOI 10.1090/S0894-0347-03-00428-4
- Alexander I. Barvinok, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res. 19 (1994), no. 4, 769–779. MR 1304623, DOI 10.1287/moor.19.4.769
- W. A. de Graaf, Classification of solvable Lie algebras, Experiment. Math. 14 (2005), no. 1, 15–25. MR 2146516
- Willem A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2007), no. 2, 640–653. MR 2303198, DOI 10.1016/j.jalgebra.2006.08.006
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
- Jan Denef, Report on Igusa’s local zeta function, Astérisque 201-203 (1991), Exp. No. 741, 359–386 (1992). Séminaire Bourbaki, Vol. 1990/91. MR 1157848
- J. Denef and F. Loeser, Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), no. 4, 705–720 (French). MR 1151541, DOI 10.1090/S0894-0347-1992-1151541-7
- The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.3), 2016. Available from http://www.sagemath.org/.
- Marcus du Sautoy, The zeta function of $\mathfrak {s}\mathfrak {l}_2(\mathbf Z)$, Forum Math. 12 (2000), no. 2, 197–221. MR 1740889, DOI 10.1515/form.2000.004
- Marcus du Sautoy and Fritz Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), no. 3, 793–833. MR 1815702, DOI 10.2307/2661355
- Marcus du Sautoy and François Loeser, Motivic zeta functions of infinite-dimensional Lie algebras, Selecta Math. (N.S.) 10 (2004), no. 2, 253–303. MR 2080122, DOI 10.1007/s00029-004-0361-y
- Marcus du Sautoy and Gareth Taylor, The zeta function of $\mathfrak {s}\mathfrak {l}_2$ and resolution of singularities, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 1, 57–73. MR 1866324, DOI 10.1017/S0305004101005369
- Marcus du Sautoy and Luke Woodward, Zeta functions of groups and rings, Lecture Notes in Mathematics, vol. 1925, Springer-Verlag, Berlin, 2008. MR 2371185, DOI 10.1007/978-3-540-74776-5
- Duong H. Dung and Christopher Voll, Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6327–6349. MR 3660223, DOI 10.1090/tran/6879
- Anton Evseev, Reduced zeta functions of Lie algebras, J. Reine Angew. Math. 633 (2009), 197–211. MR 2561201, DOI 10.1515/CRELLE.2009.065
- Shannon Ezzat, Representation growth of finitely generated torsion-free nilpotent groups: Methods and examples,\nopunct. PhD Thesis, University of Canterbury, 2012. See http://hdl.handle.net/10092/7235.
- Shannon Ezzat, Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field, J. Algebra 397 (2014), 609–624. MR 3119241, DOI 10.1016/j.jalgebra.2013.08.028
- E. S. Golod and I. R. Šafarevič, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272 (Russian). MR 0161852
- Jon González-Sánchez, Andrei Jaikin-Zapirain, and Benjamin Klopsch, The representation zeta function of a FAb compact $p$-adic Lie group vanishes at $-2$, Bull. Lond. Math. Soc. 46 (2014), no. 2, 239–244. MR 3194743, DOI 10.1112/blms/bdt090
- G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3.0, University of Kaiserslautern, 2005. Available from http://www.singular.uni-kl.de/.
- F. J. Grunewald, D. Segal, and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), no. 1, 185–223. MR 943928, DOI 10.1007/BF01393692
- Ehud Hrushovski, Ben Martin, Silvain Rideau, and Raf Cluckers, Definable equivalence relations and zeta functions of groups (2017). To appear in J. Eur. Math. Soc. (JEMS).
- Ishai Ilani, Zeta functions related to the group $\textrm {SL}_2(\mathbf Z_p)$, Israel J. Math. 109 (1999), 157–172. MR 1679595, DOI 10.1007/BF02775033
- Nicholas M. Katz, Review of $l$-adic cohomology, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 21–30. MR 1265520
- Benjamin Klopsch and Christopher Voll, Zeta functions of three-dimensional $p$-adic Lie algebras, Math. Z. 263 (2009), no. 1, 195–210. MR 2529493, DOI 10.1007/s00209-008-0416-4
- Olga Kuzmich, Graded nilpotent Lie algebras in low dimensions, Lobachevskii J. Math. 3 (1999), 147–184. Towards 100 years after Sophus Lie (Kazan, 1998). MR 1743136
- V. V. Morozov, Classification of nilpotent Lie algebras of sixth order, Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 4 (5), 161–171 (Russian). MR 0130326
- Jin Nakagawa, Orders of a quartic field, Mem. Amer. Math. Soc. 122 (1996), no. 583, viii+75. MR 1342021, DOI 10.1090/memo/0583
- Charles Nunley and Andy Magid, Simple representations of the integral Heisenberg group, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 89–96. MR 982280, DOI 10.1090/conm/082/982280
- Tobias Rossmann, Computing topological zeta functions of groups, algebras, and modules, I, Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1099–1134. MR 3349788, DOI 10.1112/plms/pdv012
- Tobias Rossmann, Computing topological zeta functions of groups, algebras, and modules, II, J. Algebra 444 (2015), 567–605. MR 3406186, DOI 10.1016/j.jalgebra.2015.07.039
- Tobias Rossmann, Topological representation zeta functions of unipotent groups, J. Algebra 448 (2016), 210–237. MR 3438311, DOI 10.1016/j.jalgebra.2015.09.050
- Tobias Rossmann, Enumerating submodules invariant under an endomorphism, Math. Ann. 368 (2017), no. 1-2, 391–417. MR 3651578, DOI 10.1007/s00208-016-1499-6
- Tobias Rossmann, Stability results for local zeta functions of groups algebras, and modules, Math. Proc. Cambridge Philos. Soc. (2017), 1–10. DOI:10.1017/S0305004117000585\nopunct.
- Tobias Rossmann, Zeta, version 0.3.2, 2017. See http://www.math.uni-bielefeld.de/~rossmann/Zeta/.
- Jean-Pierre Serre, Lectures on $N_X (p)$, Chapman & Hall/CRC Research Notes in Mathematics, vol. 11, CRC Press, Boca Raton, FL, 2012. MR 2920749
- Robert Snocken, Zeta functions of groups and rings,\nopunct. PhD Thesis, University of Southampton, 2012. See http://eprints.soton.ac.uk/id/eprint/372833.
- Louis Solomon, Zeta functions and integral representation theory, Advances in Math. 26 (1977), no. 3, 306–326. MR 460292, DOI 10.1016/0001-8708(77)90044-5
- A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $B$, Amer. J. Math. 136 (2014), no. 2, 501–550. MR 3188068, DOI 10.1353/ajm.2014.0010
- Gareth Taylor, Zeta functions of algebras and resolution of singularities, PhD Thesis, 2001.
- Christopher Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. (2) 172 (2010), no. 2, 1181–1218. MR 2680489, DOI 10.4007/annals.2010.172.1185
- Christopher Voll, Local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms, Int. Math. Res. Not. , posted on (19 August 2017)., DOI 10.1093/imrn/rnx186
- Juliette White, Zeta functions of groups,\nopunct. DPhil Thesis, University of Oxford, 2000.
- Kevin Woods and Ruriko Yoshida, Short rational generating functions and their applications to integer programming, SIAG/OPT Views-and-News 16 (2005), no. 1–2, 15–19.
- Luke Woodward, Zeta functions of groups: computer calculations and functional equations,\nopunct. DPhil Thesis, University of Oxford, 2005.
- Luke Woodward, Zeta functions of Lie rings of upper-triangular matrices, J. Lond. Math. Soc. (2) 77 (2008), no. 1, 69–82. MR 2389917, DOI 10.1112/jlms/jdm068
Additional Information
- Tobias Rossmann
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany
- Address at time of publication: Department of Mathematics, University of Auckland, Auckland, New Zealand
- Email: tobias.rossmann@gmail.com
- Received by editor(s): February 2, 2016
- Received by editor(s) in revised form: September 29, 2016
- Published electronically: December 27, 2017
- Additional Notes: This work was supported by the DFG Priority Programme “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory” (SPP 1489).
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4841-4879
- MSC (2010): Primary 11M41, 20F69, 20G30, 20F18, 20C15
- DOI: https://doi.org/10.1090/tran/7361
- MathSciNet review: 3812098