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Author : Dmitri Kaledin,Yuri Tschinkel
Publisher : IOS Press
Page : 356 pages
File Size : 55,9 Mb
Release : 2008
Category : Mathematics
ISBN : 9781586038557
"Proceedings of the NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields, Geottingen, Germany, 25 June-6 July 2007."--T.p. verso.
Author : D. Kaledin,Y. Tschinkel
Publisher : IOS Press
Page : 356 pages
File Size : 46,8 Mb
Release : 2008-06-05
Category : Mathematics
ISBN : 9781607503255
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.
Author : Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel
Publisher : Springer
Page : 261 pages
File Size : 44,5 Mb
Release : 2017-02-09
Category : Mathematics
ISBN : 9783319497631
Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.
Author : Károly Jr. Böröczky,János Kollár,Szamuely Tamas
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 52,6 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9783662051238
Exploring the connections between arithmetic and geometric properties of algebraic varieties has been the object of much fruitful study for a long time, especially in the case of curves. The aim of the Summer School and Conference on "Higher Dimensional Varieties and Rational Points" held in Budapest, Hungary during September 2001 was to bring together students and experts from the arithmetic and geometric sides of algebraic geometry in order to get a better understanding of the current problems, interactions and advances in higher dimension. The lecture series and conference lectures assembled in this volume give a comprehensive introduction to students and researchers in algebraic geometry and in related fields to the main ideas of this rapidly developing area.
Author : James William Peter Hirschfeld,Joseph Adolf Thas
Publisher : Oxford University Press, USA
Page : 432 pages
File Size : 44,7 Mb
Release : 1991
Category : Mathematics
ISBN : UOM:39015040692488
Projective spaces over a finite field, otherwise known as Galois geometries, find wide application in coding theory, algebraic geometry, design theory, graph theory, and group theory as well as being beautiful objects of study in their own right. This volume is the culmination of a three volume treatise on this subject. With its companion volumes (Projective Geometries Over Finite Fields and Finite Projective Spaces of Three Dimensions) this work will provide a major reference to the subject. It is essentially self-contained and will be an invaluable companion for research workers and post-graduate students working on these topics. The authors study three main themes: the study of algebraic varieties over finite fields, the combinatorics of Galois geometries, and the identification of various incidence structures associated with them. In particular, much attention is devoted to hermitian varieties, to Grassmannian varieties, and to polar spaces. The topics covered find application across a wide-range of topics in pure mathematics. The book will be useful to research workers and postgraduate students in combinatorics and geometry, and some computer scientists.
Author : Thomas Peternell,Joichi Miyaoka
Publisher : Birkhäuser
Page : 221 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034888936
This book is based on lecture notes of a seminar of the Deutsche Mathematiker Vereinigung held by the authors at Oberwolfach from April 2 to 8, 1995. It gives an introduction to the classification theory and geometry of higher dimensional complex-algebraic varieties, focusing on the tremendeous developments of the sub ject in the last 20 years. The work is in two parts, with each one preceeded by an introduction describing its contents in detail. Here, it will suffice to simply ex plain how the subject matter has been divided. Cum grano salis one might say that Part 1 (Miyaoka) is more concerned with the algebraic methods and Part 2 (Peternell) with the more analytic aspects though they have unavoidable overlaps because there is no clearcut distinction between the two methods. Specifically, Part 1 treats the deformation theory, existence and geometry of rational curves via characteristic p, while Part 2 is principally concerned with vanishing theorems and their geometric applications. Part I Geometry of Rational Curves on Varieties Yoichi Miyaoka RIMS Kyoto University 606-01 Kyoto Japan Introduction: Why Rational Curves? This note is based on a series of lectures given at the Mathematisches Forschungsin stitut at Oberwolfach, Germany, as a part of the DMV seminar "Mori Theory". The construction of minimal models was discussed by T.
Author : James William Peter Hirschfeld
Publisher : Oxford University Press on Demand
Page : 555 pages
File Size : 47,9 Mb
Release : 1998
Category : Law
ISBN : 0198502958
I. Introduction 1. Finite fields 2. Projective spaces and algebraic varieties II. Elementary general properties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities III. The line and the plane 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References.
Author : Gary L. Mullen,Daniel Panario
Publisher : CRC Press
Page : 1048 pages
File Size : 40,8 Mb
Release : 2013-06-17
Category : Computers
ISBN : 9781439873823
Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and
Author : Ching-Li Chai,Brian Conrad, Frans Oort
Publisher : American Mathematical Soc.
Page : 402 pages
File Size : 50,6 Mb
Release : 2013-12-19
Category : Mathematics
ISBN : 9781470410148
Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.
Author : Cristian Popescu,Karl Rubin,Alice Silverberg
Publisher : American Mathematical Soc.
Page : 517 pages
File Size : 45,5 Mb
Release : 2024-07-01
Category : Mathematics
ISBN : 9780821886984
Author : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Soc.
Page : 453 pages
File Size : 52,5 Mb
Release : 2019-07-02
Category : Coding theory
ISBN : 9781470448653
Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to local_libraryBook Catalogseveral domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.
Author : Henning Stichtenoth,Michael A. Tsfasman
Publisher : Springer
Page : 235 pages
File Size : 43,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540472674
About ten years ago, V.D. Goppa found a surprising connection between the theory of algebraic curves over a finite field and error-correcting codes. The aim of the meeting "Algebraic Geometry and Coding Theory" was to give a survey on the present state of research in this field and related topics. The proceedings contain research papers on several aspects of the theory, among them: Codes constructed from special curves and from higher-dimensional varieties, Decoding of algebraic geometric codes, Trace codes, Exponen- tial sums, Fast multiplication in finite fields, Asymptotic number of points on algebraic curves, Sphere packings.
Author : James William Peter Hirschfeld,Reader in Mathematics J W P Hirschfeld
Publisher : Oxford University Press on Demand
Page : 316 pages
File Size : 46,7 Mb
Release : 1985
Category : Mathematics
ISBN : 0198535368
This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field. It is the second and core volume of a three-volume treatise on finite projective spaces, the first volume being Projective Geometrics Over Finite Fields (OUP, 1979). The present work restricts itself to three dimensions, and considers both topics which are analogous of geometry over the complex numbers and topics that arise out of the modern theory of incidence structures. The book also examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.
Author : Zhexian Wan
Publisher : Unknown
Page : 416 pages
File Size : 42,8 Mb
Release : 1993
Category : Finite fields (Algebra).
ISBN : UOM:39015032280615
This monograph provides a survey of classical groups over finite fields. Many important results, appearing for the first time in book-form, are clearly presented. Most of them have been obtained by the author and students under his supervision.
Author : Walter Noll
Publisher : Springer Science & Business Media
Page : 403 pages
File Size : 49,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401093354
A. Audience. This treatise (consisting of the present VoU and of VoUI, to be published) is primarily intended to be a textbook for a core course in mathematics at the advanced undergraduate or the beginning graduate level. The treatise should also be useful as a textbook for selected stu dents in honors programs at the sophomore and junior level. Finally, it should be of use to theoretically inclined scientists and engineers who wish to gain a better understanding of those parts of mathemat ics that are most likely to help them gain insight into the conceptual foundations of the scientific discipline of their interest. B. Prerequisites. Before studying this treatise, a student should be familiar with the material summarized in Chapters 0 and 1 of Vol.1. Three one-semester courses in serious mathematics should be sufficient to gain such fa miliarity. The first should be an introduction to contemporary math ematics and should cover sets, families, mappings, relations, number systems, and basic algebraic structures. The second should be an in troduction to rigorous real analysis, dealing with real numbers and real sequences, and with limits, continuity, differentiation, and integration of real functions of one real variable. The third should be an intro duction to linear algebra, with emphasis on concepts rather than on computational procedures. C. Organization.