By Carlos Moreno

ISBN-10: 052134252X

ISBN-13: 9780521342520

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

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**Example text**

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).

Klk(D) is not empty, then d(D) <2g — 1. Klk consider all divisors D with co e QfK/k(D), and from among these choose one of maximal degree and denote it by (co). Suppose D and D' are two divisors associated with co. K/k(D') and D < [D,D'] and D' < [D,D'], we have that <*([£>,£>']) = degD = degD' and this implies that D = [D,D'] = D', hence the uniqueness of (co). This proves the lemma. 5 If x e K and co e Q^/jt, then (xco) = (x) + (co). Proof. K/k((co)) and x e L(-D), then xco e £%/k((co) + (x)).

Does not change under a finite extensions of k; with these cases in mind we shall simply call g the genus of the curve C. 2. e. g > 0. 5 The degree of speciality of a divisor D is the positive integer S(D) = l(D) - d(D) + 3 - 1 . 5 Pre-adeles (repartitions) Let K be the function field of the curve C with k as its field of constants. Let S be the set of all closed points on C. 6 A family {rP}PeSof elements rPe K is called a pre-adele if ordP(rP) > 0 for all P e S except possibly a finite number.