Über die Autorin bzw. den Autor

J.W.P. Hirschfeld, G. Korchmáros & F. Torres

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"Very useful both for research and in the classroom. The main reason to use this book in a classroom is to prepare students for new research in the fields of finite geometries, curves in positive characteristic in a projective space, and curves over a finite field and their applications to coding theory. I think researchers will quote it for a long time."--Edoardo Ballico, University of Trento

"This book is a self-contained guide to the theory of algebraic curves over a finite field, one that leads readers to various recent results in this and related areas. Personally I was attracted by the rich examples explained in this book."--Masaaki Homma, Kanagawa University

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"Very useful both for research and in the classroom. The main reason to use this book in a classroom is to prepare students for new research in the fields of finite geometries, curves in positive characteristic in a projective space, and curves over a finite field and their applications to coding theory. I think researchers will quote it for a long time."--Edoardo Ballico, University of Trento

"This book is a self-contained guide to the theory of algebraic curves over a finite field, one that leads readers to various recent results in this and related areas. Personally I was attracted by the rich examples explained in this book."--Masaaki Homma, Kanagawa University

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Algebraic Curves over a Finite Field

Princeton University Press

Chapter One

In this chapter, basic facts about curves are presented. The exposition also highlights some of the peculiarities that occur for positive characteristic, such as the existence of strange curves, that is, curves whose tangent lines at non- singular points have a point in common.

1.1 BASIC DEFINITIONS

Over the real numbers, R, consider the parabola F given by F = Y - [X.sup.2]; its points form, as in Figure 1.1, the set

{(t, [t.sup.2])| t [member of] R}.

However, there are two other types of points associated with F, namely, (a) those at infinity and (b) those with coordinates in BLDBLD, the algebraic closure of R. For example, regarding (b), the line with equation y + 1=0 meets F in the two points (i, -1), (-i, -1), where [i.sup.2] = -1. Regarding (a), if F is homogenised to [F.sup.*] = [X.sub.0][X.sub.2] - X.sup.2.sub.1], with X = [X.sub.1]/[X.sub.0], Y = [X.sub.2]/[X.sub.0], then the line with equation [X.sub.0] =0 meets the corresponding projective curve [F.sup.*] at the point (0, 0, 1).

All these ideas need to be considered for a general curve and a general field. First, some notation and fundamental definitions for the spaces that appear are explained.

Definition 1.1 (i) For a field K, let [K.sup.n] = {([x.sub.1], [x.sub.2], ..., [x.sub.n])| [x.sub.i] [member of] K}, the n-fold Cartesian product of K.

(ii) Let V (n, K) be n-dimensional vector space over K, which may be regarded as ([K.sup.n] , +, .), where, for [x.sub.i], [y.sub.i], [lambda] [member of] K,

([x.sub.1], [x.sub.2], ..., [x.sub.n]) + ([y.sub.1], [y.sub.2], ..., [y.sub.n]) = ([x.sub.1] + [y.sub.1], [x.sub.2] + [y.sub.2], ..., [x.sub.n] + [y.sub.n]), [lambda]([x.sub.1], [x.sub.2], ..., [x.sub.n]) = ([lambda][x.sub.1], [lambda][x.sub.2], ..., [lambda][x.sub.n]).

(iii) The affine plane AG(2, K)= [A.sup.2](K) is a pair (P, L), where P = {P =(x, y)| x, y [element of K}, L = {l = aX + bY + c | a, b, c [element of] K, (a, b) [not equal to] (0, 0)},

and a point P =(x, y) lies on a line l = aX + bY + c if ax + by + c =0.

(iv) More generally, affine space of n-dimensions is AG(n, K) = [A.sup.n](K) with points x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and r-dimensional subspaces x + S, for r-dimensional subspaces S of V (n, K).

(v) The projective plane PG(2, K)= [P.sup.2](K) is a pair (P, L), where

P = {P = (x, y, z) = ([lambda]x, [lambda]y, [lambda]z) | (x, y, z) [element of] [K.sup.3]\{(0, 0, 0)}, [lambda] [element of] K\{0}}, L = {l = aX + bY + cZ = [lambda]aX + [lambda]bY + [lambda]cZ | a, b, c, [lambda] [element of] K, (a, b, c) [not equal to] (0, 0, 0), [lambda] [not equal to] 0}, and a point P =(x, y, z) lies on a line l = aX + bY + cZ if ax+by+cz = 0.

(vi) More generally, projective space of n-dimensions is PG(n, K)= [P.sup.n](K) with points,

x = ([x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n]) = ([lambda][x.sub.0], [lambda][x.sub.1], [lambda][x.sub.2], ..., [lambda][x.sub.n]), ([x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n]) [not equal to] (0, 0, 0, ..., 0), [lambda] [not equal to] 0, and r-dimensional subspaces S, for (r + 1)-dimensional subspaces S of V (n + 1, K).

In each type of space, it is important to consider the structure-preserving transformations.

Definition 1.2 (i) A linear transformation T : V (n, K) [right arrow] V(n, K), is given as follows:

T(x)= [x.sup.' where [sup.t]x' = A [sup.t]x for a suitable non-singular matrix A,

with x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]), [x.sup.'] = ([x.sup'.sub.1], [x.sup.'.sub.2], ..., [x.sup.'.sub.n]), and [sup.t]x the transpose of x. The linear transformations of V (n, K)constitute the the general linear group GL(n, K).

A semilinear transformation T : V (n, K) [right arrow] V (n, K), is given as follows:

T(x)= x', where [sup.t]x' = A [sup.t][sigma](x)for a suitable non-singular matrix A, with [sigma](x) = ([sigma]([x.sub.1]), [sigma]([x.sub.2]), ..., [sigma]([x.sub.n])) for some automorphism [sigma] of K.

The semilinear transformations of V (n, K) constitute its general semilinear group [GAMMA] L(n, K).

(ii) An affine transformation S : AG(n, K) [right arrow] AG(n, K) is given as follows: S(x) = x' = T (x) + b,

where T is a linear transformation and b = ([b.sub.1], [b.sub.2], ..., [b.sub.n]) The affine transformations of AG(n, K) constitute its affine group AGL(n, K).

An affine collineation S : AG(n, K) [right arrow] AG(n, K) is given as follows: S(x) = x' = T ([sigma](x)) + b,

with T as above and [sigma](x) as in (i).

(iii) A projectivity T : PG(n, K) [right arrow] PG(n, K)is given as follows:

T(x) = x', where [sup.t]x' = A [sup.t]x,

with

x = ([x.sub.0], [x.sub.1], ..., [x.sub.n]), x' = ([x.sup.'.sub.0], [x'.sub.1], ..., [x'.sub.n]),

and A a suitable non-singular matrix. It is also called a projective transformation or linear collineation. The projectivities of PG(n, K) constitute its projective general linear group PGL(n + 1, K).

A collineation T : PG(n, K) [right arrow] PG(n, K) is given as follows:

T(x) = x', where [sup.t]x' = A [sup.t][sigma](x),

with A as above and [sigma](x) = ([sigma]([x.sub.0]), ..., [sigma]([x.sub.n])).

The collineations of PG(n, K) constitute its projective semilinear group P[GAMMA]L(n + 1, K).

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