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 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 outcomes for the estimation of exponential sums in a single variable; Goppa's thought of error-correcting codes created from linear structures on algebraic curves; there's additionally a brand new evidence of the TsfasmanSHVladutSHZink theorem. the must haves had to stick with 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 idea of error-correcting codes also will take advantage of learning this paintings.

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**Additional info for Algebraic Curves over Finite Fields**

**Sample text**

For notational convenience we let Oi /t (D) = Homk(A/(A(D) + K),k); we observe that if the divisors D, D' satisfy D < D', then &Klk(D') £ ClsK/k(D). Hence we can view the space of pseudo-differentials on K as a projective limit ci"m = lim arKlk(D). 8 Let 0 denote the zero divisor of K; the space QsK/k(&) is called the space of pseudo-differentials of the first kind. °Klk(D) = S(D). e. dimkn*Klk(<9) = g. 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 if this were not the case then we could represent 1 in the form 1 = j ^ y , + ••• + xhyh, with x, e mA, y{ e AY (1 <. i < h). Each yt would belong to some ring A(i) e &'. For each pair (i,j) one of the rings Am, AU) would contain the other; since there are only a finite number of rings A(i), it follows that they would all be contained in one of them, say A{k). But then we would have 1 = £ ? ^ e mA • A(k\ therefore mA • A{k) = Alk\ which is impossible since Aik) e &'. ¥, and hence 3F is inductive.

Any element x e RQ has a representation of the form d x = X a/*; + *'> J=l with x' e WIQ. , d and all P e S, we obtain ordP(ux,) + ordP(D + Q) > 0; hence MX, e L(D + Q)s. An element x e L(D + Q)s satisfies ord e (X) + ord e (D + Q) > 0 and hence ordQ(xu~x) = ordQ(x) — ordQ(u) = ord e (u) + ord e (D + Q) > 0, that is to say xu'1 e RQ; therefore we have d XM"1 = X aJxJ + x'> j=l with aj e k and x' e mQ. , d and all PeS - {Q} it follows that ord P (x') > 0 and therefore ord P (ux') + ordP(D) = ordP(wx') + ord P (D + Q) > 0 for all P e S - {Q}.

### Algebraic Curves over Finite Fields by Carlos Moreno

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