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Higher-Dimensional Geometry Over Finite Fields

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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.

The papers included in this publication 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 of at least 2.

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Author(s): Y. Tschinkel, Y. Tschinkel
Series: NATO Science for Peace and Security Series. Sub-Series D: In
Edition: illustrated edition
Publisher: IOS Press
Year: 2008

Language: English
Pages: 356

Title page......Page 1
Preface......Page 5
Contents......Page 7
Finite Field Experiments......Page 9
K3 Surfaces of Picard Rank One Which Are Double Covers of the Projective Plane......Page 71
Beilinson Conjectures in the Non-Commutative Setting......Page 86
Looking for Rational Curves on Cubic Hypersurfaces......Page 100
Abelian Varieties over Finite Fields......Page 131
How to Obtain Global Information from Computations over Finite Fields......Page 197
Geometry of Shimura Varieties of Hodge Type over Finite Fields......Page 205
Lectures on Zeta Functions over Finite Fields......Page 252
De Rham Cohomology of Varieties over Fields of Positive Characteristic......Page 277
Homomorphisms of Abelian Varieties over Finite Fields......Page 323
Author Index......Page 353