structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80, and their non-existence has been proven for dimensions above 163,264. Classification For indefinite lattices, the classification is easy to describe. Write Rm,n for the m + n dimensional vector space Rm+n with the inner product of (a1, ..., am+n) and (b1, ..., bm+n) given by In Rm,n there is one odd indefinite unimodular lattice up to isomorphism, denoted by Im,n, which is given by all vectors (a1,...,am+n) in Rm,n with all the ai integers. There are no indefinite even unimodular lattices unless m − n is divisible by 8, in which case there is a unique example up to isomorphism, denoted by IIm,n. This is given by all vectors (a1,...,am+n) in Rm,n such that either all the ai are integers or they are all integers plus 1/2, and their sum is even. The lattice II8,0 is the same as the E8 lattice. Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example In,0 in each dimension n less than 8, and two examples (I8,0 and II8,0) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the Smith-Minkowski-Siegel mass formula implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32. In some sense unimodular lattices up to dimension 9 are controlled by E8, and up to dimension 25 they are controlled by the Leech lattice, and this accounts for their unusually good behavior in these dimensions. For example, the Dynkin diagram of the norm-2 vectors of unimodular lattices in dimension up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by the Leech lattice. Even positive definite unimodular lattice exist only in dimensions divisible by 8. There is one in dimension 8 (the E8 lattice), two in dimension 16 (E82 and II16,0),