On even unimodular positive definite quadratic lattices of rank 32. (English) Zbl 0564.10016

Let \(\Gamma_{32}\) be the genus of all equivalence classes of positive definite even unimodular lattices of rank 32. Let L be a representative of an element in \(\Gamma_{32}\). A vector x in L is called 2m-vector if x satisfies \((x,x)=2m\) for some positive integer m, where (x,x) is the metric for L. We let \({\mathcal L}_{2m}(L)\) denote the sublattice of L generated by all 2m-vectors in L. It is known that \({\mathcal L}_ 2(L)\) is an orthogonal sum of the so-called root lattices of types \(A_ n\) (n\(\geq 1)\), \(D_ n\) (n\(\geq 4)\), \(E_ n\) \((n=6,7,8)\). Two representatives \(L_ 1\) and \(L_ 2\) in two classes in \(\Gamma_{32}\) are called of the same 2-type if \({\mathcal L}_ 2(L_ 1)\) is isomorphic to \({\mathcal L}_ 2(L_ 2).\)
In this paper, the following results are proved: (1) There are at most 367 157 different 2-types in \(\Gamma_{32}\) (theorem 1). (2) For any representative L of \(\Gamma_{32}\), it holds that rank \({\mathcal L}_ 4(L)=32\) (theorem 2). - In the end, some problems are proposed in connection with the above results.


11E12 Quadratic forms over global rings and fields
11F27 Theta series; Weil representation; theta correspondences
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
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