2001/01/01 by Ichiro Shimada · 3 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Finite Group Theory Research
paper · doi:10.1112/s0024611500012673
Let X be the Fermat hypersurface of dimension 2m and of degree q+1 defined over an algebraically closed field of characteristic p>0, where q is a power of p, and let NLm(X) be the free abelian group of numerical equivalence classes of linear subspaces of dimension m contained in X. By the intersection form, we regard NLm(X) as a lattice. Investigating the configuration of these linear subspaces, we show that the rank of NLm(X) is equal to the 2mth Betti number of X, that the intersection form multiplied by (−1)m is positive definite on the primitive part of NLm(X), and that the discriminant of NLm(X) is a power of p. Let Lm(X) be the primitive part of NLm(X) equipped with the intersection form multiplied by (−1)m. In the case p=q=2, the lattice Lm(X) is described in terms of certain codes associated with the unitary geometry over F2. Since L1(X) is isomorphic to the root lattice of type E6, the series of lattices Lm(X) can be considered as a generalization of E6. The lattice L2(X) is isomorphic to the laminated lattice of rank 22. This isomorphism explains Conway's identification ·222 ≅ PSU(6,2) geometrically. The lattice L3(X) is of discriminant 216 · 3, minimal norm 8, and kissing number 109421928. 2000 Mathematics Subject Classification: 14C25, 11H31, 51D25.