By Carlos Moreno

During this tract, Professor Moreno develops the speculation of algebraic curves over finite fields, their zeta and L-functions, and, for the 1st time, the idea of algebraic geometric Goppa codes on algebraic curves. one of the functions thought of are: the matter of counting the variety of strategies of equations over finite fields; Bombieri's evidence of the Reimann speculation for functionality fields, with effects for the estimation of exponential sums in a single variable; Goppa's concept of error-correcting codes produced from linear structures on algebraic curves; there's additionally a brand new facts of the TsfasmanSHVladutSHZink theorem. the must haves had to persist with this e-book are few, and it may be used for graduate classes for arithmetic scholars. electric engineers who have to comprehend the fashionable advancements within the thought of error-correcting codes also will reap the benefits of learning this paintings.

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When the field of constants k is algebraically closed, the group C1O(C), which is usually denoted by Jac(C), has the structure of an algebraic variety with an abelian 36 The Riemann-Roch theorem group law. e. those whose function fields are abelian extensions of K, are parametrized by the subgroups C1O(C). 2 we obtain the following corollary. 2 If x is a non-constant element in Kx, then there is an integer H such that for all integers m. Proof. 2 the inequality JV(r+l) = \K:k[-) \(t + 1) ^ l((x'+X)- If we put m = s + t with m> s, then /((*"%) - d((xm)J > (1 - W ( 4 ) = -A*.

In fact if x e L(-D), then x cannot be a constant because ord P (x) > ord,,( + D) > 0 for some closed point P. The function x must be constant for otherwise it would have a pole at some point Q, thus contradicting the assumption that ordP(x) > ordP(D) > 0 for all P. This shows that L(-D) = {0} and hence by (ii) dimtL(£>') = d(D') + d(D). e. the divisor with ord P (0) = 0 for all P, then L(O) = k and l((9) = 1. 3 Principal divisors and the group of divisor classes In this section we introduce the notion of a principal divisor and show that its degree is zero.

S; hence dim t L(D + Q)S/L(D)S < d. , ad in k, the element y = YJ=I a,*;" does not belong to L(D)S. i=\ aj*'j ^ ®> an<* hence ordgfyu"1) = 0 or equivalently ord 0 >' + ord 0 (D + Q) = ordQ(y) + ord 0 (D) + 1 = 0. On the other hand if y e L(D)S then ord Q y + ord e (D) > 0, which is impossible. 1. 2 The vector space L(D) The statement of the Riemann-Roch theorem refers to a vector space which generalizes L(D)S and whose definition we now present. 2 Let S be the set of all closed points of C. For D a divisor in Div(C) we put = {xe Kx: ordp(x) + ordP(D) > 0 for all P e S} and L{D) is a vector space over k and its dimension is denoted by = dim t L(D).

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