## Algebraic Curves over Finite Fields by Carlos Moreno

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 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 idea of error-correcting codes comprised of linear structures on algebraic curves; there's additionally a brand new facts of the TsfasmanSHVladutSHZink theorem. the necessities had to stick to this publication are few, and it may be used for graduate classes for arithmetic scholars. electric engineers who have to comprehend the fashionable advancements within the conception of error-correcting codes also will take advantage of learning this paintings.

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We shall now prove that the vector space A/A(D) + K is finite dimensional over k. Observe first that if D < D' and r e A(D), then ordP(r) + ord P D' = ord P r -f ord P D + ord P D' - ordPZ) > 0 and hence A(D) £ A(D'). Let D e Div(C) and r e A. Define an integral divisor Dr = Y a(P)P, where a(P) = 0 if ord P (r) > 0 and a(P) = — ord P (r) otherwise. Clearly ord P (r) + a(P) > 0 and hence r e A(Dr). d. e. D' = Y, max(ordP(£>),ordP(Dr))P. p 40 The Riemann-Roch theorem Clearly Dr < D' and hence r e A(Dr) c A(D').

Zhe Kx such that det(vi(zj)) # 0. To show this we shall construct these elements step by step. For zx we take any element in Kx with vx(zx) ¥^0. ,n)e Q": £ rtv,(*i) = oj =ft- 1. =i r,i;,(zk+1) / 0. , r») e Q*: £ wfa) = 0,1 < ;

Proof. Since ord P (x) = 0 for all xe k, we may assume that x is not a constant, (ii) clearly follows from (i). As for (i), it suffices to show that there is only a finite number of closed points P with ordi,(x) > 0; the same argument will give that ord P (x" 1 ) > 0 for a finite number of closed points P. Let S be a finite set of closed points for which ordP(x) > 0. If 0 is the zero divisor and D = Y,'PesOTdp(x)P, where ^ V e s denotes that ordP(x) is set equal to zero for all P not in S, then dim* L(Z))s/L(0)s = d(D) = X / P ordp(x).