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We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds' method.

In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that logλd, where λd is the minimal Salem number of degree d, is realized in characteristic 5 if and only if d⩽22 is even and d≠18. In the complex projective setting we settle the case of entropy logλ12, left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, that is, preserves a chamber of the positive cone.

We classify pairs (X, G) consisting of a complex K3 surface X and a finite group G Aut(X) such that the subgroup G s G consisting of symplectic automorphisms is among the 11 maximal symplectic ones as classified by Mukai.Résumé. -Nous classifions les paires (X, G) formées d'une surface K3 complexe X et d'un groupe fini G Aut(X) pour lesquelles le sous-groupe G s G des automorphismes symplectiques appartient aux 11 sous-groupes symplectiques maximaux classifiés par Mukai.

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves hermitian lattices over number fields.