## 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.

### MSC:

 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)

### Keywords:

root lattices; different 2-types; rank
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### References:

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