Algebraic Curves over a Finite Field
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A01=Fernando Torres
A01=Gabor Korchmaros
A01=J. W. P. Hirschfeld
Affine plane
Affine space
Affine variety
Algebraic closure
Algebraic curve
Algebraic equation
Algebraic extension
Algebraic function
Algebraic geometry
Algebraic integer
Algebraic number
Algebraic number field
Algebraic number theory
Algebraic variety
Algebraically closed field
Author_Fernando Torres
Author_Gabor Korchmaros
Author_J. W. P. Hirschfeld
Automorphism
Birational invariant
Category=PBMW
Classification theorem
Clifford's theorem
Combinatorics
Cyclotomic polynomial
Degeneracy (mathematics)
Divisor
Divisor (algebraic geometry)
Dual curve
Elliptic curve
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equation
Finite field
Finite geometry
Function (mathematics)
Function field
Galois extension
Galois theory
Gauss map
Generic point
Geometry
Hurwitz's theorem
Hyperelliptic curve
Hyperplane
Intersection number (graph theory)
J-invariant
Line at infinity
Linear map
Mathematical induction
Mathematics
Menelaus' theorem
Modular curve
Number theory
Parity (mathematics)
Permutation group
Plane curve
Point at infinity
Polar curve
Polynomial
Projective plane
Projective space
Quadratic transformation
Resolution of singularities
Riemann hypothesis
Scalar multiplication
Separable extension
Separable polynomial
Sign (mathematics)
Subgroup
Sylow theorems
Theorem
Transcendence degree
Valuation ring
Variable (mathematics)
Vector space
Product details
- ISBN 9780691096797
- Weight: 1162g
- Dimensions: 152 x 235mm
- Publication Date: 23 Mar 2008
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Hardback
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This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stohr and Voloch.
The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
J.W.P. Hirschfeld is professor emeritus of mathematics at the University of Sussex. His books include "Projective Geometries over Finite Fields". G. Korchmaros is professor of mathematics at the University of Basilicata in Italy. F. Torres is professor of mathematics at the University of Campinas in Brazil.
