Modular lattices from finite projective planes
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 269-279.

Using the geometry of the projective plane over the finite field 𝔽 q , we construct a Hermitian Lorentzian lattice L q of dimension (q 2 +q+2) defined over a certain number ring 𝒪 that depends on q. We show that infinitely many of these lattices are p-modular, that is, pL q ' =L q , where p is some prime in 𝒪 such that |p| 2 =q.

The Lorentzian lattices L q sometimes lead to construction of interesting positive definite lattices. In particular, if q3mod4 is a rational prime such that (q 2 +q+1) is norm of some element in [-q], then we find a 2q(q+1) dimensional even unimodular positive definite integer lattice M q such that Aut(M q )PGL(3,𝔽 q ). We find that M 3 is the Leech lattice.

En utilisant la géométrie du plan projectif sur un corps fini 𝔽 q , nous construisons un réseau hermitien de type Lorentz L q de dimension (q 2 +q+2) defini sur un certain anneau d’entiers 𝒪 dépendant de q. Nous montrons qu’une infinité de ces réseaux sont p-modulaires, c’est-à-dire que pL q ' =L q , où p est un premier de 𝒪 tel que |p| 2 =q.

Les réseaux lorentziens L q mènent parfois à la construction de réseaux définis positifs intéressants. En particulier, si q3mod4 est tel que (q 2 +q+1) est la norme d’un élément de [-q], alors nous obtenons un réseau entier unimodulaire M q défini positif et de dimension paire 2q(q+1) tel que Aut(M q )PGL(3,𝔽 q ). Nous prouvons que M 3 est le réseau de Leech.

DOI: 10.5802/jtnb.867
Classification: 11H56, 51E20, 11E12, 11E39
Tathagata Basak 1

1 Department of Mathematics Iowa State University Ames, IA 50011
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Tathagata Basak. Modular lattices from finite projective planes. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 269-279. doi : 10.5802/jtnb.867. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.867/

[1] D. Allcock, The Leech lattice and complex hyperbolic reflections. Invent. Math. 140 (2000), 283–301. | MR | Zbl

[2] D. Allcock, A monstrous proposal. Groups and Symmetries: From neolithic Scots to John McKay (2009), AMS and Centre de Recherches Math士atiques. arXiv:math/0606043. | MR | Zbl

[3] D. Allcock, On the Y 555 complex reflection group. J. Alg. 322 (2009), 1454–1465 | MR | Zbl

[4] R. Bacher and B. Venkov, Lattices and association schemes: A unimodular example without roots in dimension 28. Ann. Inst. Fourier, Grenoble. 45, 5 (1995), 1163–1176. | EuDML | Numdam | MR | Zbl

[5] T. Basak, The complex Lorentzian Leech lattice and the bimonster. J. Alg. 309, no. 1 (2007), 32–56. | MR | Zbl

[6] T. Basak, Reflection group of the quaternionic Lorentzian Leech lattice. J. Alg. 309, no. 1 (2007), 57–68. | MR | Zbl

[7] T. Basak, The complex Lorentzian Leech lattice and the bimonster (II). preprint (2012), arXiv:0811.0062, submitted. | MR

[8] A. I. Bondal, Invariant lattices in Lie algebras of type A p-1 (Russian). Vestnik Moskov. Univ. Ser. I Mat. Mekh. 93, no. 1 (1986), 52–54. | MR | Zbl

[9] J. H. Conway, S. P. Norton and L. H. Soicher, The bimonster, the group Y 555 , and the projective plane of order 3. “ Computers in Algebra" (M. C. Tangara, Ed.), Lecture Notes in Pure and Applied Mathematics, No 111, Dekker, New York, (1988), 27–50. | MR | Zbl

[10] J. H. Conway and C. S. Simons, 26 Implies the Bimonster. J. Alg. 235 (2001), 805–814. | MR | Zbl

[11] J. H. Conway, and N. J. A. Sloane, Sphere Packings, Lattices and Groups 3rd Ed. Springer-Verlag, 1998. | Zbl

[12] A. Fröhlich and M.J. Taylor, Algebraic number theory. Cambridge University Press, 1991. | MR | Zbl

[13] R. L. Graham and J. Macwilliams, On the number of information symbols in the difference set cyclic codes. The Bell System Technical Journal, Vol XLV, No. 7, 1966. | MR | Zbl

[14] A. A. Ivanov, A geometric characterization of the monster. Groups, Combinatorics and Geometry, edited by M. Liebeck and J. Saxl, London Mathematical Society Lecture Note Series, No. 165, Cambridge Univ. Press, (1992), 46–62. | MR | Zbl

[15] A. A. Ivanov, Y–groups via transitive extension. J. Alg. 218 (1999) 412–435. | MR | Zbl

[16] A. I. Kostrikin and H. T. Pham, Orthogonal decompositions and integral lattices. Walter de Gruyter, 1994. | MR | Zbl

[17] G. Nebe and K. Schindelar, S-extremal strongly modular lattices. J. Théor. Nombres Bordeaux, 19 no. 3, (2007), 68–701. | Numdam | MR | Zbl

[18] J. P. Serre, A course in Arithmetic. Springer-Verlag, 1973. | MR | Zbl

[19] J. Singer, A theorem in finite projective geometry and some applications to number theory. Trans. A. M. S. 43, No. 3 (1938), 377–385. | MR

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