Topics in Algebraic Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

An algebraic surface of type K3 is a 2-dimensional analog of an elliptic curve. It is characterized by the property that its tangent bundle is not trivial but the first Chern class is trivial. Its group of algebraic automorphisms 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 most important and oldest problems in mathematics. A symmetry group of a Riemann surface or an algebraic curve is now well understood. Much less is known about symmetries of higher dimensional algebraic varieties. The principal inverstigator proposes such study for a class of algebraic surfaces known as K3 surfaces which are two-dimensional analogs of elliptic curves. The symmetry groups of K3 surfaces are related to symmetry 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.