By Carlos Moreno

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 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 results for the estimation of exponential sums in a single variable; Goppa's conception of error-correcting codes made out of linear structures on algebraic curves; there's additionally a brand new evidence of the TsfasmanSHVladutSHZink theorem. the must haves had to persist 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 conception of error-correcting codes also will reap the benefits of learning this paintings.

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**Sample text**

If D is a divisor in % then N(2) = l(D). 1 Let Dbea divisor in Di v(C). Then the number of positive divisors in Div(C) equivalent to D is (qHD) - \)/{q - 1). Proof. As in Chapter 2, we consider the set & (D) = {feKx: (f) + D> 0}. The set L(D) = ¥(D) u {0} is a vector space over k of dimension /(/)), containing q'(D) elements; the set if(D) itself contains ql(D) — 1 elements. Observe that to each function / e if(D) there corresponds a divisor D' = D + (/) which is positive and in the same divisor class as D.

14 (1986)). For a brief survey of the classical literature it may be worthwhile to consult J. Dieudonne's History of Algebraic Geometry (Wadsworth Adv. Books and Software (1985)). 1 Divisors Throughout this chapter K will denote a field of algebraic functions of one variable with k as the exact field of constants. For applications to number theory and coding theory it will be useful to specialize k as the finite field ¥q or its algebraic closure F, = (J* =1 F,n. 1 in Chapter 1; with this dictionary in mind, we shall refer interchangeably to the discrete valuation rings of K as the closed points of C.

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*. ) > -/*• This proves the corollary. 4 The Riemann theorem In this section we establish that part of the Riemann-Roch theorem which is originally due to Riemann. This will be used later to prove the full theorem. 3 (Riemann) Let xbea non-constant function in K and define an integer g by 1 - g = min {l((xm)J - d((xm)J}. me Z Then for any divisor D in Div(C) we have l(D) > d(D) + 1 - g; in particular the integer g is independent of x.