Topics in Algebraic Geometry

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An algebraic surface of type K3 is a 2-dimensional analog of an ellipticcurve. It is characterized by the property that its tangent bundle is nottrivial but the first Chern class is trivial. Its group of algebraicautomorphisms is a discrete group sometimes infinite sometimes finite and its structure is closely related to the structure of the orthogonal group of the the Picard group of divisor classes equipped with the intersection product. The structure of the automorphism group of a complex K3 surface is well understood thanks to the availability of trascendental methods based on the study of the integration of a holomorphic 2-form on the surface over transcendental cycles. No such methods are available in the case when the characteristic of the ground field is positive. In the proposal the principal investigator outlines several new approaches to the study of automorphism groups of K3 surfaces over such fields. Some of them based on the study of possible automorphisms of finite order which will allow to compute the character of the group in its representation on l-adic cohomology. Other approaches use the relationship between the Picard lattice and the 24-dimensional Leech lattice. The principal investigator will also study some applications to coding theory and cryptology related to K3 surfaces over a finite field.The study of symmetries of mathematical structures is one of the mostimportant and oldest problems in mathematics. A symmetry group of aRiemann surface or an algebraic curve is now well understood. Much less is known about symmetries of higher dimensional algebraic varieties. Theprincipal inverstigator proposes such study for a class of algebraicsurfaces known as K3 surfaces which are two-dimensional analogs ofelliptic curves. The symmetry groups of K3 surfaces are related tosymmetry of other objects, for example lattices in hyperbolic spaces and convex polyhedra. Many known abstract infinite and finite groups admit a beautiful realization as symmetry groups of K3 surfaces. Applications of symmetry groups of elliptic curves over finite fields to coding theory and cryptology is well known. It is expected that the knowledge of symmetry groups of K3 surfaces over finite field will find new applications to these theories.