Algebraic Curves over Finite Fields [Paperback]

Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.The theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves are developed in this text. Electrical engineers as well as mathematics students will find the material of interest.The theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves are developed in this text. Electrical engineers as well as mathematics students will find the material of interest.In this tract, Professor Moreno develops the theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves. Among the applications considered are: the problem of counting the number of solutions of equations over finite fields; Bombieri's proof of the Reimann hypothesis for function fields, with consequences for the estimation of exponential sums in one variable; Goppa's theory of error-correcting codes constructed from linear systems on algebraic curves; there is also a new proof of the TsfasmanSHVladutSHZink theorem. The prerequisites needed to follow this book are few, and it can be used for graduate courses for mathematics students. Electrical engineers who need to understand the modern developments in the theory of error-correcting codes will also benefit from studying this work.1. Algebraic curves and function fields; 2. The RiemannRoch theorem; 3. Zeta functions; 4. Applications to exponential sums and zeta functions; 5. Applications to coding theory; Bibliography. ...well written and has many interesting exercises. It could be used as a textbook for a nice course addressed to students with some background in algebra or number theory. José Felipe lÃ9