AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The reflective Lorentzian lattices of rank 3
About this Title
Daniel Allcock, Department of Mathematics, U.T. Austin
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 220, Number 1033
ISBNs: 978-0-8218-6911-6 (print); 978-0-8218-9203-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00648-4
Published electronically: March 1, 2012
Keywords: Lorentzian lattice,
Weyl group,
Coxeter group,
Vinberg’s algorithm
MSC: Primary 11H56; Secondary 20F55, 22E40
Table of Contents
Chapters
- Introduction
- 1. Background
- 2. The Classification Theorem
- 3. The Reflective Lattices
Abstract
We classify all the symmetric integer bilinear forms of signature $(2,1)$ whose isometry groups are generated up to finite index by reflections. There are 8,595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin’s enumeration of the strongly square-free cases. Our technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg’s algorithm and a “method of bijections”. We also correct a minor error in Conway and Sloane’s definition of their canonical $2$-adic symbol.- Allcock, Daniel, Unabridged table of reflective Lorentzian lattices of rank $3$, available at arXiv:1111.1264.
- Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200, DOI 10.1016/0021-8693(87)90245-6
- Richard E. Borcherds, The monster Lie algebra, Adv. Math. 83 (1990), no. 1, 30–47. MR 1069386, DOI 10.1016/0001-8708(90)90067-W
- Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
- Richard E. Borcherds, Reflection groups of Lorentzian lattices, Duke Math. J. 104 (2000), no. 2, 319–366. MR 1773561, DOI 10.1215/S0012-7094-00-10424-3
- Richard E. Borcherds, Coxeter groups, Lorentzian lattices, and $K3$ surfaces, Internat. Math. Res. Notices 19 (1998), 1011–1031. MR 1654763, DOI 10.1155/S1073792898000609
- J. H. Conway, The automorphism group of the $26$-dimensional even unimodular Lorentzian lattice, J. Algebra 80 (1983), no. 1, 159–163. MR 690711, DOI 10.1016/0021-8693(83)90025-X
- J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15 (1982), no. 1, 83–94. MR 666350, DOI 10.1016/0022-314X(82)90084-1
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447
- H. S. M. Coxeter and G. J. Whitrow, World-structure and non-Euclidean honeycombs, Proc. Roy. Soc. London Ser. A 201 (1950), 417–437. MR 41576, DOI 10.1098/rspa.1950.0070
- Frank Esselmann, Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe, J. Number Theory 61 (1996), no. 1, 103–144 (German, with English summary). MR 1418323, DOI 10.1006/jnth.1996.0141
- V. A. Gritsenko and V. V. Nikulin, On the classification of Lorentzian Kac-Moody algebras, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 79–138 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 5, 921–979. MR 1992083, DOI 10.1070/RM2002v057n05ABEH000553
- V. V. Nikulin, On the arithmetic groups generated by reflections in Lobačevskiĭ spaces, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 637–669, 719–720 (Russian). MR 582161
- V. V. Nikulin, On the classification of arithmetic groups generated by reflections in Lobachevskiĭ spaces, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 113–142, 240 (Russian). MR 607579
- V. V. Nikulin, $K3$ surfaces with interesting groups of automorphisms, J. Math. Sci. (New York) 95 (1999), no. 1, 2028–2048. Algebraic geometry, 8. MR 1708598, DOI 10.1007/BF02169159
- V. V. Nikulin, On the classification of hyperbolic root systems of rank three, Tr. Mat. Inst. Steklova 230 (2000), 256 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. Math. 3(230) (2000), 1–241. MR 1802343
- V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by $2$-reflections, Dokl. Akad. Nauk SSSR 248 (1979), no. 6, 1307–1309 (Russian). MR 556762
- V. V. Nikulin, $K3$ surfaces with a finite group of automorphisms and a Picard group of rank three, Trudy Mat. Inst. Steklov. 165 (1984), 119–142 (Russian). Algebraic geometry and its applications. MR 752938
- PARI/GP, version 2.4.3, Bordeaux, 2008, http://pari.math.u-bordeaux.fr
- I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
- Scharlau, R., On the classification of arithmetic groups on hyperbolic 3-space, preprint Bielefeld 1989.
- Rudolf Scharlau and Britta Blaschke, Reflective integral lattices, J. Algebra 181 (1996), no. 3, 934–961. MR 1386588, DOI 10.1006/jabr.1996.0155
- Rudolf Scharlau and Claudia Walhorn, Integral lattices and hyperbolic reflection groups, Astérisque 209 (1992), 15–16, 279–291. Journées Arithmétiques, 1991 (Geneva). MR 1211022
- È. B. Vinberg, Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1072–1112 (Russian). MR 0302779
- È. B. Vinberg, The groups of units of certain quadratic forms, Mat. Sb. (N.S.) 87(129) (1972), 18–36 (Russian). MR 0295193
- È. B. Vinberg, The two most algebraic $K3$ surfaces, Math. Ann. 265 (1983), no. 1, 1–21. MR 719348, DOI 10.1007/BF01456933
- È. B. Vinberg, Classification of 2-reflective hyperbolic lattices of rank 4, Tr. Mosk. Mat. Obs. 68 (2007), 44–76 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2007), 39–66. MR 2429266, DOI 10.1090/S0077-1554-07-00160-4
- È. B. Vinberg and I. M. Kaplinskaja, The groups $O_{18,1}(Z)$ and $O_{19,1}(Z)$, Dokl. Akad. Nauk SSSR 238 (1978), no. 6, 1273–1275 (Russian). MR 0476640
- Walhorn, Claudia, Arithmetische Spiegelungsgruppen auf dem 4-dim\discretionary{-}-en\discretionary{-}-sion\discretionary{-}-alen hyperbolische Raum, dissertation Bielefeld 1993.
- G. L. Watson, Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. (3) 12 (1962), 577–587. MR 142512, DOI 10.1112/plms/s3-12.1.577
- G. L. Watson, Transformations of a quadratic form which do not increase the class-number. II, Acta Arith. 27 (1975), 171–189. MR 364100, DOI 10.4064/aa-27-1-171-189