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."
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