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 recommendations of equations over finite fields; Bombieri's facts of the Reimann speculation for functionality fields, with effects 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 facts of the TsfasmanSHVladutSHZink theorem. the necessities had to keep on with this ebook 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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**Extra resources for Algebraic Curves over Finite Fields**

**Sample text**

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).

Are linearly independent over k. We want to show that the ed elements tiUj, 1 < i < e, 1

Let k be the algebraic closure of the finite field F, and define a group H = {X e kx: kq*1 = 1}. Drienfeld has studied the action of SL2(F,) by linear change of coordinates and of H by homothety on the affine irreducible curve ^ = {(x,y)ek2:xyq~x'>y= 1}. Show how to extend this action to the non-singular projective curve Show that over the field F,2 the curve % can be written as xt+l + yq+l + zq+l on which the unitary group l/3(F,) acts naturally. What relation is there between these two group actions?