Read e-book online Algebraic Curves over Finite Fields PDF

By Carlos Moreno

ISBN-10: 052134252X

ISBN-13: 9780521342520

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 purposes thought of are: the matter of counting the variety of options 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 thought of error-correcting codes comprised of linear platforms on algebraic curves; there's additionally a brand new facts of the TsfasmanSHVladutSHZink theorem. the must haves had to persist with this booklet are few, and it may be used for graduate classes for arithmetic scholars. electric engineers who have to comprehend the trendy advancements within the thought of error-correcting codes also will make the most of learning this paintings.

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We first show that for a fixed positive integer d0 there can be at most a finite number of closed points P on C with bounded degree d{P) < d0. Let x be a non-constant function in K* and let D = (x)w be its divisor of poles. ln fact if Q $ supp(D), then ordQ(x) > 0 and hence x is an element of the local ring RQ of Q. 2 The zeta functions of curves 51 corresponds to a monic irreducible polynomial p(x) in the polynomial ring A = /c[x] and we also have deg(P) = deg(p(x)) < deg(0; hence if d(Q) < d0, then d(P) < d0.

K/k = 1 imply that the composite map &Klk ^ Div(C) - Div(C)/Diva(C) = given by co -»(co) mod Divfl(C), sends ilsK/k into a unique divisor class. e. the divisor class of (co) for some pseudo-differential co. In the following the canonical class will be denoted by W and any divisor in W will be oenoted by w. We are now ready to state the main result of this chapter. 5 (Riemann-Roch) / / D is a divisor in Div(C) and w is a divisor in the canonical class W, then = d(D) +l-g + l(w-D). Proof. sK/k(D) given by x -> xco.

If D, D' are two divisors satisfying D < D', then L(D)S £ L(D')S and dimk L(D')S/L(D)S = d(D's) - d(Ds). Proof. Since ord P D < ordP£>' for all P e S, the inequality ordp(/) + ordp(D') = ord F (/) + ordP(D) + ord P (D') - ordP(D) > 0, holds for all fe L(D)S and all P e S; hence L(D)S s L(D')S. To establish the second assertion we observe that HP)s — L(Ds)s and L(D')S = L(D'S); we may thus suppose that D = Ds and D' = D's. Clearly it suffices to prove that for any closed point Q in S we have In fact, if D' = D + Qx + •• + Qh, then the inclusions 30 The Riemann-Roch theorem L(D)S £ L(D + Qx)s £ • • • £ L(D + Q, + • • • + Qh)s = L(D') S yield dim, L(Z)')S/L(Z))S = £ dim, L(D + Q, + •• • + Qds/UD + g , + ••• + Q ^ J s ; i= l the second claim will follow from d(D') - d(D) = £ d(Qt).

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Algebraic Curves over Finite Fields by Carlos Moreno


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