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Many Rational Points : Coding Theory and Algebraic Geometry - Norman E. Hurt

Many Rational Points

Coding Theory and Algebraic Geometry

Hardcover

Published: 30th November 2003
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2 Triangle Groups: An Introduction 279 3 Elementary Shimura Curves 281 4 Examples of Shimura Curves 282 5 Congruence Zeta Functions 283 6 Diophantine Properties of Shimura Curves 284 7 Klein Quartic 285 8 Supersingular Points 289 Towers of Elkies 9 289 7. CRYPTOGRAPHY AND APPLICATIONS 291 1 Introduction 291 Discrete Logarithm Problem 2 291 Curves for Public-Key Cryptosystems 3 295 Hyperelliptic Curve Cryptosystems 4 297 CM-Method 5 299 6 Cryptographic Exponent 300 7 Constructive Descent 302 8 Gaudry and Harley Algorithm 306 9 Picard Jacobians 307 Drinfeld Module Based Public Key Cryptosystems 10 308 11 Drinfeld Modules and One Way Functions 308 12 Shimura's Map 309 13 Modular Jacobians of Genus 2 Curves 310 Modular Jacobian Surfaces 14 312 15 Modular Curves of Genus Two 313 16 Hecke Operators 314 8. REFERENCES 317 345 Index Xll Preface The history of counting points on curves over finite fields is very ex- tensive, starting with the work of Gauss in 1801 and continuing with the work of Artin, Schmidt, Hasse and Weil in their study of curves and the related zeta functions Zx(t), where m Zx(t) = exp (2: N t ) m m 2': 1 m with N = #X(F qm). If X is a curve of genus g, Weil's conjectures m state that L(t) Zx(t) = (1 - t)(l - qt) where L(t) = rr~!l (1 - O'.

From the reviews:

"This book gives a nice overview of background and recent results on curves over finite fields. ... The main advantage of this book is that it provides a huge bibliography and takes into account even very recent results which are so far only presented at conferences or in preprints. So it serves well to get an update on recent results for the experienced reader and links to the original results for more details." (Tanja Lange, Zentralblatt MATH, Vol. 1072 (23), 2005)

Preface
Abelian Varietiesp. 1
Refined Boundsp. 127
Codes and Curvesp. 155
Deligne-Lusztig Spacesp. 207
Drinfeld Modulesp. 259
Shimura Curvesp. 279
Cryptography and Applicationsp. 291
Referencesp. 317
Indexp. 345
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9781402017667
ISBN-10: 1402017669
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 346
Published: 30th November 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.52
Weight (kg): 0.8