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.

Show description

Read Online or Download Algebraic Curves over Finite Fields PDF

Best algebraic geometry books

Equidistribution in Number Theory, An Introduction

Written for graduate scholars and researchers alike, this set of lectures offers a based creation to the concept that of equidistribution in quantity thought. this idea is of growing to be value in lots of components, together with cryptography, zeros of L-functions, Heegner issues, top quantity idea, the speculation of quadratic kinds, and the mathematics points of quantum chaos.

Undergraduate algebraic geometry

Algebraic geometry is, primarily, the learn of the answer of equations and occupies a primary place in natural arithmetic. With the minimal of must haves, Dr. Reid introduces the reader to the fundamental suggestions of algebraic geometry, together with: airplane conics, cubics and the crowd legislations, affine and projective forms, and nonsingularity and measurement.

Constructible Sets in Real Geometry

This booklet offers a scientific and unified record at the minimum description of constructible units. It starts off at a really easy point (almost undergraduate) and leads as much as state of the art effects, lots of that are released in publication shape for the first actual time. The ebook comprises quite a few examples, sixty three figures and every bankruptcy ends with a bit containing old notes.

Algebraic geometry for scientists and engineers

This ebook, in accordance with lectures offered in classes on algebraic geometry taught through the writer at Purdue college, is meant for engineers and scientists (especially computing device scientists), in addition to graduate scholars and complicated undergraduates in arithmetic. as well as offering a concrete or algorithmic method of algebraic geometry, the writer additionally makes an attempt to inspire and clarify its hyperlink to extra smooth algebraic geometry in keeping with summary algebra.

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.

Download PDF sample

Rated 4.09 of 5 – based on 35 votes