# Algebraic Curves over Finite Fields by Carlos Moreno

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 suggestions 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 conception of error-correcting codes constituted of linear platforms on algebraic curves; there's additionally a brand new facts of the TsfasmanSHVladutSHZink theorem. the necessities had to persist with this publication are few, and it may be used for graduate classes for arithmetic scholars. electric engineers who have to comprehend the trendy advancements within the idea of error-correcting codes also will make the most of learning this paintings.

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Additional info for Algebraic Curves over Finite Fields

Example text

The unit element of K is also the unit element of A; since A contains an isomorphic image of K, we may consider the ring of pre-adeles as a vector space over K and therefore also as a vector space over k. In the following we shall consider A mainly as a vector space over k. If r = {rP} is a pre-adele and P is a closed point, then we set ordP(r) = ord P (r P ), where rP is the P-component of r. Observe that if r is a principal pre-adele, then ordP(r) agrees with the earlier definition; we shall refer to ordP(r) as the order of the pre-adele r at P.

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. Let B be a maximal ring in J^. We claim that B is a valuation ring of K. Consider the set it is clearly closed under multiplication, and the localization of B at S is the ring Bs= {Ps-l:peB,seS}. 20 Algebraic curves and function fields We have B s B s . We show first that 1 £ mA • Bs. - e « ^ , A e B, st e S, (1 ^ i ±s fc). /, with aj e ««4, ft' e B, and we have which is impossible.

For applications to number theory and coding theory it will be useful to specialize k as the finite field ¥q or its algebraic closure F, = (J* =1 F,n. 1 in Chapter 1; with this dictionary in mind, we shall refer interchangeably to the discrete valuation rings of K as the closed points of C. 1 A divisor of K is an element of the free Abelian group generated by the set of closed points of C. The group of divisors of K is denoted by Div(C); the group operation will be written additively. Thus any element D in Div(C) has the form D = £ ord P (D)P, p where the sum is taken over all closed points P of C, and the coefficients ordp(Z)) are integers all of which are zero except for a finite number.