Ebook: Higher-Dimensional Geometry over Finite Fields

Number systems based on a finite collection of symbols, such as the 0s and 1s of computer circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields.

These are the proceedings of the NATO Advanced Study Institute “Higher-dimensional geometry over finite fields” held at the University of Göttingen in June–July 2007. They introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension at least 2.

Dmitry Kaledin, Yuri Tschinkel

We show how to use experiments over finite fields to gain information about the solution set of polynomial equations in characteristic zero.

We construct examples of K3 surfaces over which are of degree 2 and the geometric Picard rank of which is equal to 1. We construct, particularly, examples in the form w²=det M where M is a symmetric (3×3)-matrix of ternary quadratic forms or a symmetric (6×6)-matrix of ternary linear forms. Our method is based on reduction modulo p for p=3 and 5.

We discuss a p-adic version of Beilinson's conjecture and its relationship with noncommutative geometry.

We study rational points and rational curves on varieties over finite fields. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.

A. Weil proved that the geometric Frobenius π=F^a of an abelian variety over a finite field with q=p^a elements has absolute value for every embedding. T. Honda and J. Tate showed that gives a bijection between the set of isogeny classes of simple abelian varieties over F_q and the set of conjugacy classes of q-Weil numbers.

This is an extended version of the talk I gave at the summer school in Göttingen in July 2007. We discuss the Mordell-Weil Sieve and some applications.

We present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli interpretations, their uniqueness, their smoothness, their properness, and basic stratifications of their special fibres.

We give an introduction to zeta functions over finite fields, focusing on moment zeta functions and zeta functions of affine toric hypersurfaces.

These are the elaborated notes of two talks given at the Summer School in Göttingen on Higher-Dimensional Geometry over Finite Fields. We study the De Rham cohomology of smooth and proper varieties over fields of positive characteristic in case that the Hodge spectral sequence degenerates. The De Rham cohomology carries the structure of a so-called F-zip. We explain two classifications of F-zips, one stems from representation theory of algebras and the other one uses algebraic groups and their compactifications. We show how this second classification can be extended if the De Rham cohomology is endowed with a symplectic or a symmetric pairing. Throughout we illustrate the theory via the examples of (polarized) abelian varieties and (polarized) K3-surfaces.

We give a proof of Tate's theorems on homomorphisms of abelian varieties over finite fields and the corresponding ℓ-divisible groups.