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

Gross, Benedict H.; McMullen, Curtis T.

J. Algebra 257, No. 2, 265-290 (2002).

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 \(S(-1)S(1)=-1\).

Reviewer: Helmut Koch (Berlin)

MSC:

11E12

Quadratic forms over global rings and fields

Keywords:

characteristic polynomials of automorphisms; even unimodular lattices; monic polynomials; Salem polynomial; complex K3 surface

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