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 ideas 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 made out of linear platforms on algebraic curves; there's additionally a brand new evidence of the TsfasmanSHVladutSHZink theorem. the must haves had to stick to this booklet 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 make the most of learning this paintings.

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We now want to show that Q^ /t (also known as the dualizing module) may also be viewed as a vector space over K. In fact think of K as the subset of principal pre-adeles in A and for x e K and u> e £%/jt(D) put xft)(r) = co(xr) for any r e A/(A (D) + K). K/k(D - (x)). The following properties are immediate consequences of the definitions: for x, y e K and co, co' e i^ /jk we have (i) (xy)a> = x(yco), (ii) (x + y)a> = xa> + yu> (iii) x(w + a)') = xo) + xw'. This shows that Q^ /t is a vector space over K.

If we let C1(C) = Div(C)/Diva(C) and C1O(C) = Divo(C)/Diva(C) then we obtain the exact sequence 0 -> C1O(C) -> C1(C) -> dT -»0, which is of fundamental importance in the study of the zeta function of the curve C. It will be shown later that indeed 8=1. 2 For any integer d, the number ofdivisor classes in C1(C) of degree d is independent of d and is equal to the cardinality of C1O(C). Proof. 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.

If x, y e Kx, then (x><) = (x) + (y) and the set Divo(C) = {(x): x e K*} 34 The Riemann-Roch theorem of all principal divisors is a subgroup of Div(C). 1 shows that d((x)0) < IK : k(x)l and d((x)J < [K : fc(x)]. 2 If x is a non-constant element in Kx, then Proof. e. y satisfies a monic polynomial equation ym + fm-i W / " " 1 + • • • + fo{x) = o, with fj€ k [x], then P is a pole of y only if it is a pole of x; equivalently, P appears in (y)x only if it appears in {x)x. Observe that if P does not appear in (x)x, then ordP(x) > 0 and therefore /m-l \ mordp(y) = ordP(ym) = o r d J X &x)yJ \j=0 ^ ) min {jordp(y),0} OsjSm-l = ; 0 ord P (y), with 0

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Algebraic Curves over Finite Fields by Carlos Moreno

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