Algebraic Curves over Finite Fields by Carlos Moreno

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By Carlos Moreno

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 strategies of equations over finite fields; Bombieri's evidence of the Reimann speculation for functionality fields, with effects for the estimation of exponential sums in a single variable; Goppa's idea of error-correcting codes made from linear structures on algebraic curves; there's additionally a brand new evidence of the TsfasmanSHVladutSHZink theorem. the must haves had to persist with this e-book 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 reap the benefits of learning this paintings.

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To deal with the divisor classes of degree < g — 1 we can first translate them by a divisor with positive degree and then apply the previous argument. In particular, the number of divisor classes of any degree is equal to the number of divisor classes of degree 0, which is simply the cardinality of C1O(C). This proves the lemma. Remark C1O(Q is the analogue of the class group of an algebraic number field; thus when convenient we shall refer to the cardinality of C1O(C) as the class number of the curve C and denote it by h{C).

5 Pre-adeles (repartitions) Let K be the function field of the curve C with k as its field of constants. Let S be the set of all closed points on C. 6 A family {rP}PeSof elements rPe K is called a pre-adele if ordP(rP) > 0 for all P e S except possibly a finite number. The element rP is called the P-th component of the pre-adele r = {rP}PeS. e. a subset of the cartesian product f^Pes^Remark If we were to allow the elements rP to lie in the completion of K with respect to the valuation ord P , we would obtain the ordinary notion of adele.

5 For any two divisors D, D' e Div(C), we have where [D, D'] is t/ie least common multiple of D and D'. Proof. sKlk(D)nQisKlk{D'). To verify the inclusion in the other direction we consider an element r e A ([£>,£']) and observe that ord P (r)> -max{ord P (D),ordp(D')} holds for all P. Write r = r' + r" where the P component of r' and r" are given by rP = rP for all P with ord P D > ord P D' and rP = 0 and rP = rP for all P with ordp D < ord P D'. ). This decomposition of r implies w(r) = co(r') +

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