Hostname: page-component-857557d7f7-bkbbk Total loading time: 0 Render date: 2025-12-02T09:40:01.022Z Has data issue: false hasContentIssue false
  • English
  • Français

VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF $GL_N(\mathbb{Z} )$ FOR $N\geqslant 8$ AND THE TRIVIALITY OF $K_8(\mathbb{Z} )$

Part of: Computational number theory Connections with homological algebra and category theory Higher algebraic $K$-theory Discontinuous groups and automorphic forms Geometry of numbers Homology and cohomology theories

Published online by Cambridge University Press:  24 November 2025

Mathieu Dutour Sikirić ,
Philippe Elbaz-Vincent Open the ORCID record for Philippe Elbaz-Vincent [Opens in a new window] ,
Alexander Kupers  and
Jacques Martinet
Mathieu Dutour Sikirić
Affiliation:
Rudjer Bošković Institute , Zagreb, Croatia Current address: MSM Programming, 10450 Jastrebarsko, Croatia (mathieu.dutour@gmail.com)
Philippe Elbaz-Vincent*
Affiliation:
Institut Fourier, Université Grenoble Alpes , France
Alexander Kupers
Affiliation:
Department of Mathematics, University of Toronto , Canada (a.kupers@utoronto.ca)
Jacques Martinet
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux , France (jacques.martinet@math.cnrs.fr)
*
(philippe.elbaz-vincent@math.cnrs.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We enumerate the low-dimensional cells in the Voronoi cell complexes attached to the modular groups $\mathit {SL}_N(\mathbb{Z} )$ and $\mathit {GL}_N(\mathbb{Z} )$ for $N=8,9,10,11$, using quotient sublattice techniques for $N=8,9$ and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that $K_8(\mathbb{Z} ) = 0$. We deduce from it new knowledge on the Kummer-Vandiver conjecture.

Keywords

Perfect formsVoronoi complexgroup cohomologymodular groupsSteinberg modulesK-theory of integerswell-rounded lattices

MSC classification

Primary: 11H55: Quadratic forms (reduction theory, extreme forms, etc.) 19D50: Computations of higher $K$-theory of rings 20J06: Cohomology of groups
Secondary: 11F75: Cohomology of arithmetic groups 11F06: Structure of modular groups and generalizations; arithmetic groups 11Y99: None of the above, but in this section 55N91: Equivariant homology and cohomology

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arlettaz, D (1991) The Hurewicz homomorphism in algebraic $K$ -theory. J. Pure Appl. Algebra 71(1), 1–12.10.1016/0022-4049(91)90036-2CrossRefGoogle Scholar
Bacher, R (2018) On the number of perfect lattices. J. Théor. Nombres Bordeaux 30(3), 917–945.10.5802/jtnb.1057CrossRefGoogle Scholar
Barnes, ES (1957) The complete enumeration of extreme senary forms. Philos. Trans. Roy. Soc. London. Ser. A. 249, 461–506.Google Scholar
A-M, Bergé and Martinet, J (2009) On perfection relations in lattices. Contemp. Math. 493, 29–49.Google Scholar
Borel, A (1975), 1974 Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7, 235–272.10.24033/asens.1269CrossRefGoogle Scholar
Borel, A and J-P, Serre (1973) Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491. Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault.10.1007/BF02566134CrossRefGoogle Scholar
Bosma, W, Cannon, J, and Playoust, C (1997) The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3–4), 235–265. Computational algebra and number theory (London, 1993).10.1006/jsco.1996.0125CrossRefGoogle Scholar
Brown, KS (1994) Cohomology of Groups, volume 87 of Graduate Texts in Mathematics. New York: Springer-Verlag. Corrected reprint of the 1982 original.Google Scholar
Brück, B, Miller, J, Patzt, P, Sroka, R and Wilson, JCH (2024) On the codimension-two cohomology of SL ${}_n(\mathbb{Z})$ . Adv. Math. 451, https://doi.org/10.1016/j.aim.2024.109795.CrossRefGoogle Scholar
Buhler, JP and Harvey, D (2011) Irregular primes to 163 million. Math. Comp. 80(276), 2435–2435.10.1090/S0025-5718-2011-02461-0CrossRefGoogle Scholar
Church, T, Farb, B and Putman, A (2014) A stability conjecture for the unstable cohomology of $\operatorname{SL}_n(\mathbb{Z})$ , mapping class groups, and $\operatorname{Aut}(F_n)$ , in Algebraic topology: applications and new directions. Stanford symposium on algebraic topology: applications and new directions. Contemp. Math. 620, 55–70.10.1090/conm/620/12366CrossRefGoogle Scholar
Church, T and Putman, A (March 2017) The codimension-one cohomology of SL ${}_n(\mathbb{Z})$ . Geom. Topol. 21(2), 999–1032.10.2140/gt.2017.21.999CrossRefGoogle Scholar
Cohen, H (1993) A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Berlin: Springer-Verlag.10.1007/978-3-662-02945-9CrossRefGoogle Scholar
Dutour Sikirić, M, Hulek, K and Schürmann, A (2015) Smoothness and singularities of the perfect form and the second Voronoi compactification of $\mathcal{A}_g$ . Algebr. Geom. 2(5), 642–653.10.14231/AG-2015-027CrossRefGoogle Scholar
Dutour Sikirić, M, Schürmann, A and Vallentin, F (2007) Classification of eight-dimensional perfect forms. Electron. Res. Announc. Amer. Math. Soc. 13, 21–32.10.1090/S1079-6762-07-00171-0CrossRefGoogle Scholar
Dutour Sikirić, M and van Woerden, WPJ (2025) The lattice packing problem in dimension 9 by voronoi’s algorithm. Preprint at arXiv:2508.20719.Google Scholar
Dwyer, WG and Friedlander, EM (1985) Algebraic and etale $K$ -theory. Trans. Amer. Math. Soc. 292(1), 247–280.Google Scholar
Elbaz-Vincent, P (2016) ‘Perfect forms of rank $\le 8$ , triviality of ${K}_8(\mathbb{Z})$ and the Kummer/Vandiver conjecture’, in Coulangeon, R, Gross, BH and Nebe, G, editors, Lattices and Applications in Number Theory, Oberwolfach Rep. 13, 87–154. doi: 10.4171/OWR/2016/3.CrossRefGoogle Scholar
Elbaz-Vincent, P, Gangl, H and Soulé, C (2013) Perfect forms, K-theory and the cohomology of modular groups. Adv. Math. 245, 587–624.10.1016/j.aim.2013.06.014CrossRefGoogle Scholar
Farrell, FT (1977/78) An extension of Tate cohomology to a class of infinite groups. J. Pure Appl. Algebra 10(2), 153–161.10.1016/0022-4049(77)90018-4CrossRefGoogle Scholar
Fousse, L, Hanrot, G, Lefèvre, V, Pélissier, P and Zimmermann, P (2007) MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Software 33(2), Art. 13, 15.10.1145/1236463.1236468CrossRefGoogle Scholar
Fukuda, K (2016). cdd. https://www.inf.ethz.ch/personal/fukudak/cdd_home.Google Scholar
Gauss, C-F (1840) Untersuchungen über die eigenschaften der positiven ternären quadratischen formen von ludwig august seeber. J. Reine Angew. Math. 20, 312–320.Google Scholar
The GLPK Group (2017). GLPK. https://www.gnu.org/software/glpk.Google Scholar
The PARI Group (2023). PARI/GP version 2.15.4, http://pari.math.u-bordeaux.fr/.Google Scholar
Grushevsky, S, Hulek, K and Tommasi, O (2018) Stable betti numbers of (partial) toroidal compactifications of the moduli space of abelian varieties. In Proceedings in honour of Nigel Hitchin’s 70th birthday, Volume II, page 581–610. Oxford University Press. With an appendix by Mathieu Dutour Sikirić.Google Scholar
Guennebaud, G, Jacob, B, et al. (2010) Eigen v3. http://eigen.tuxfamily.org.Google Scholar
Hart, W, Harvey, D and Ong, W (2017) Irregular primes to two billion. Math. Comp. 86(308), 3031–3049.10.1090/mcom/3211CrossRefGoogle Scholar
Jaquet-Chiffelle, D-O (1993) Énumération complète des classes de formes parfaites en dimension 7. Ann. Inst. Fourier (Grenoble) 43(1), 21–55.10.5802/aif.1320CrossRefGoogle Scholar
Keller, W, Martinet, J and Schürmann, A (2012) On classifying Minkowskian sublattices. Math. Comp. 81(278), 1063–1092. With an appendix by Mathieu Dutour Sikirić.10.1090/S0025-5718-2011-02528-7CrossRefGoogle Scholar
Korkin, AN and Zolotarev, EI (1872) Sur les formes quadratiques positives quaternaires. Math. Ann. 5(1), 581–583.10.1007/BF01442912CrossRefGoogle Scholar
Korkin, AN and Zolotarev, EI (1877) Sur les formes quadratiques positives. Math. Ann. 11(1), 242–292.10.1007/BF01442667CrossRefGoogle Scholar
Kupers, A (2017) A short proof that ${K}_8(\mathbb{Z})\cong 0$ . https://www.utsc.utoronto.ca/people/kupers/wp-content/uploads/sites/50/2021/01/k8zshorter.pdf. Google Scholar
Kurihara, M (1992) Some remarks on conjectures about cyclotomic fields and $K$ -groups of $\mathbb{Z}$ . Compos. Math. 81(2), 223–236.Google Scholar
Kuzmanovich, J and Pavlichenkov, A (2002) Finite groups of matrices whose entries are integers. Amer. Math. Monthly 109(2), 173–186.10.1080/00029890.2002.11919850CrossRefGoogle Scholar
Lagrange, J-L (1770) Démonstration d’un théorème d’arithmétique. Nouv. Mém. Acad. Berlin, in Oeuvre de Lagrange III 20, 189–201.Google Scholar
Lee, R and Szczarba, RH (1976) On the homology and cohomology of congruences subgroups. Invent. Math. 33, 15–53.10.1007/BF01425503CrossRefGoogle Scholar
Lee, R and Szczarba, RH (1978) On the torsion in ${K}_4(\mathbb{Z})$ and ${K}_5(\mathbb{Z})$ . Duke Math. J. 45(1), 101–129.10.1215/S0012-7094-78-04508-8CrossRefGoogle Scholar
Martinet, J (2001) Sur l’indice d’un sous-réseau (avec un appendice de Christian Batut). Martinet, Jacques (ed.), Euclidean lattices, spherical designs and modular forms. On the works of Boris Venkov. Genève: L’Enseignement Mathématique. Monogr. Enseign. Math. 37, 163–211.Google Scholar
Martinet, J (2003) Perfect Lattices in Euclidean Spaces, volume 327 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag.10.1007/978-3-662-05167-2CrossRefGoogle Scholar
Mazur, B and Wiles, A (1984) Class fields of abelian extensions of $\mathbb{Q}$ . I nvent. Math. 76(2), 179–330.Google Scholar
Miller, J, Nagpal, R and Patzt, P (2020) Stability in the high-dimensional cohomology of congruence subgroups. Compos. Math. 156(4), 822–861.10.1112/S0010437X20007046CrossRefGoogle Scholar
Quillen, D (1973) Finite generation of the groups ${K}_i$ of rings of algebraic integers. Algebraic $K$ -theory, I: Higher $K$ -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash, 1972), pages 179–198. Lecture Notes in Math., Vol. 341.Google Scholar
Quillen, D (1973) Higher algebraic $K$ -theory. I. Algebraic $K$ -theory, I: Higher $K$ -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85–147. Lecture Notes in Math., Vol. 341.Google Scholar
Schrijver, A (1986) Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Chichester: John Wiley & Sons, Ltd. A Wiley-Interscience Publication.Google Scholar
Schürmann, A (2009) Enumerating perfect forms. In Quadratic Forms—Algebra, Arithmetic, and Geometry, volume 493 of Contemp. Math., page 359–377. Providence, RI: American Mathematical Society.10.1090/conm/493/09679CrossRefGoogle Scholar
Soulé, C (1978) The cohomology of SL ${}_3(\mathbb{Z})$ . Topology 17(1), 1–22.10.1016/0040-9383(78)90009-5CrossRefGoogle Scholar
Soulé, C (1979) K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math. 55, 251–296.10.1007/BF01406843CrossRefGoogle Scholar
Soulé, C (1999) Perfect forms and the Vandiver conjecture. J. Reine Angew. Math. 517, 209–221.10.1515/crll.1999.095CrossRefGoogle Scholar
Soulé, C (2000) On the $3$ -torsion in ${K}_4(\mathbb{Z})$ . Topology 39(2), 259–265.10.1016/S0040-9383(99)00006-3CrossRefGoogle Scholar
van Woerden, WPJ (2020) An upper bound on the number of perfect quadratic forms. Adv. Math. 365, 107031.10.1016/j.aim.2020.107031CrossRefGoogle Scholar
Voronoi, G (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques 1: Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math. 133(1), 97–178.10.1515/crll.1908.133.97CrossRefGoogle Scholar
Weibel, CA (2013) The $K$ -Book, volume 145 of Graduate Studies in Mathematics. Providence, RI: American Mathematical Society. An introduction to algebraic $K$ -theory.Google Scholar