Read e-book online Algebraic Curves over Finite Fields PDF

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

Show description

Read Online or Download Algebraic Curves over Finite Fields PDF

Similar algebraic geometry books

Get An Invitation to Algebraic Geometry PDF

It is a description of the underlying ideas of algebraic geometry, a few of its very important advancements within the 20th century, and a few of the issues that occupy its practitioners this day. it really is meant for the operating or the aspiring mathematician who's surprising with algebraic geometry yet needs to achieve an appreciation of its foundations and its targets with at least must haves.

Lectures on Algebraic Statistics (Oberwolfach Seminars) by Mathias Drton PDF

How does an algebraic geometer learning secant kinds additional the certainty of speculation checks in records? Why could a statistician engaged on issue research increase open difficulties approximately determinantal types? Connections of this sort are on the center of the hot box of "algebraic statistics".

Read e-book online Fundamentals of the Theory of Operator Algebras, Vol. 2: PDF

This paintings and basics of the speculation of Operator Algebras. quantity I, undemanding idea current an creation to useful research and the preliminary basics of $C^*$- and von Neumann algebra concept in a kind appropriate for either intermediate graduate classes and self-study. The authors offer a transparent account of the introductory parts of this crucial and technically tough topic.

Additional resources for Algebraic Curves over Finite Fields

Sample text

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. Let x be a non-constant function in K* and let D = (x)w be its divisor of poles. ln fact if Q $ supp(D), then ordQ(x) > 0 and hence x is an element of the local ring RQ of Q. 2 The zeta functions of curves 51 corresponds to a monic irreducible polynomial p(x) in the polynomial ring A = /c[x] and we also have deg(P) = deg(p(x)) < deg(0; hence if d(Q) < d0, then d(P) < d0.

K/k = 1 imply that the composite map &Klk ^ Div(C) - Div(C)/Diva(C) = given by co -»(co) mod Divfl(C), sends ilsK/k into a unique divisor class. e. the divisor class of (co) for some pseudo-differential co. In the following the canonical class will be denoted by W and any divisor in W will be oenoted by w. We are now ready to state the main result of this chapter. 5 (Riemann-Roch) / / D is a divisor in Div(C) and w is a divisor in the canonical class W, then = d(D) +l-g + l(w-D). Proof. sK/k(D) given by x -> xco.

If D, D' are two divisors satisfying D < D', then L(D)S £ L(D')S and dimk L(D')S/L(D)S = d(D's) - d(Ds). Proof. Since ord P D < ordP£>' for all P e S, the inequality ordp(/) + ordp(D') = ord F (/) + ordP(D) + ord P (D') - ordP(D) > 0, holds for all fe L(D)S and all P e S; hence L(D)S s L(D')S. To establish the second assertion we observe that HP)s — L(Ds)s and L(D')S = L(D'S); we may thus suppose that D = Ds and D' = D's. Clearly it suffices to prove that for any closed point Q in S we have In fact, if D' = D + Qx + •• + Qh, then the inclusions 30 The Riemann-Roch theorem L(D)S £ L(D + Qx)s £ • • • £ L(D + Q, + • • • + Qh)s = L(D') S yield dim, L(Z)')S/L(Z))S = £ dim, L(D + Q, + •• • + Qds/UD + g , + ••• + Q ^ J s ; i= l the second claim will follow from d(D') - d(D) = £ d(Qt).

Download PDF sample

Algebraic Curves over Finite Fields by Carlos Moreno

by Thomas

Rated 4.15 of 5 – based on 28 votes