Algebraic Cycles and Even Unimodular Lattices

The number (up to isomorphism) of positive‐definite, even, unimodular lattices of rank 8 r grows rapidly with r . However, Bannai [ 1 ] has shown that, when counted according to weight, those with non‐trivial automorphisms make up a fraction of the whole, which goes rapidly to zero as r →∞. Therefore it is of some interest to produce families of positive‐definite, even, unimodular lattices with large automorphism groups and unbounded ranks. Suppose that G is a finite group and V is an irreducible Q[ G ]‐module such that V ⊗R is still irreducible. Then, as observed by Gross [ 8 ], the space of G ‐invariant symmetric bilinear forms on V is one‐dimensional and is necessarily generated by a positive‐definite form, unique up to scaling by non‐zero positive rationals. Thompson [ 23 ] showed that, if V is also irreducible mod p for all primes p , then it contains an invariant lattice (unique up to scaling) which is even and unimodular with appropriate scaling of the quadratic form. Examples arising in this manner are the E 8 ‐lattice of rank 8, the Leech lattice of rank 24 and the Thompson–Smith lattice of rank 248. Gow [ 6 ] has also constructed some examples associated with the basic spin representations of 2 A n and 2 S n .