Topics in Algebraic Geometry

K3型代数曲面是椭圆曲线的二维模拟曲面。它的特征在于它的切丛不是平凡的，但第一个Chern类是平凡的。它的代数自同构群是离散群，有时是无限的，有时是有限的，它的结构与配备交积的Picard类的群正交群的结构密切相关。复数K3曲面的自同构群的结构已被很好地理解，这要归功于基于超越圈上全纯2-形式积分研究的先验方法。在地面场特性为正的情况下，没有这样的方法可用。在该提案中，主要研究者概述了几种新的方法来研究这些域上K3曲面的自同构群。其中一些是基于对可能的有限阶自同构的研究，这将允许计算群在L上同调上同调表示中的特征标。其他方法使用Picard晶格和24维Leech晶格之间的关系。主要研究人员还将研究有限域上与K3曲面有关的编码理论和密码学的一些应用。数学结构对称性的研究是数学中最重要和最古老的问题之一。黎曼曲面或代数曲线的对称群现在已经很好地理解了。关于高维代数簇的对称性，我们知之甚少。主逆子建议对一类称为K3曲面的代数曲面进行这样的研究，K3曲面是椭圆曲线的二维模拟。K3曲面的对称群与其他对象的对称性有关，例如双曲空间中的格子和凸多面体。许多已知的抽象无限群和有限群被认为是K3曲面的对称群。有限域上椭圆曲线的对称群在编码理论和密码学中的应用是众所周知的。有限域上K3曲面对称群的知识有望在这些理论中找到新的应用。

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.