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Automorphisms of even unimodular lattices and unramified Salem numbers. (English) Zbl 1022.11016

The authors study the characteristic polynomials of automorphisms of even, indefinite, unimodular lattices with signature \((p,q)\). It is well known that such lattices exist if and only if \(p\equiv 0 \bmod 8\) and that they are unique up to isomorphism. The authors are interested in the question of which monic polynomials arise in this way. In particular they show that this is the case for any Salem polynomial \(S(x)\) of degree \(2n=p+q\) satisfying \(S(-1)S(1)= (-1)^n\).
The authors apply their results in the case of the lattice \(H^2(X,\mathbb{Z})\) of a complex \(K3\) surface \(X\). They show the existence of such surfaces \(X\) and automorphisms \(f:X\to X\) such that the corresponding characteristic polynomial is equal to a given Salem polynomial \(S(X)\) of degree 22 with \(S(-1)S(1)=-1\).

MSC:

11E12 Quadratic forms over global rings and fields
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References:

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