Algebraic Curves over a Finite Field (Princeton Series in Applied Mathematics): 20 - Hardcover

Hirschfeld, J. W. P.; Korchmaros, G; Torres, F; Korchmáros, G.; Torres, F.

Hardcover

ISBN 10: 0691096791 ISBN 13: 9780691096797

Publisher: Princeton University Press, 2008

Synopsis

This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves.

The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

About this title

About the Author

J.W.P. Hirschfeld is professor emeritus of mathematics at the University of Sussex. His books include "Projective Geometries over Finite Fields". G. Korchmaros is professor of mathematics at the University of Basilicata in Italy. F. Torres is professor of mathematics at the University of Campinas in Brazil.

From the Back Cover

"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

From the Inside Flap

"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

Excerpt. İ Reprinted by permission. All rights reserved.

Algebraic Curves over a Finite Field

By J.W.P. Hirschfeld G. Korchmros F. Torres

Princeton University Press

Copyright İ 2008 Princeton University Press
All right reserved.
ISBN: 978-0-691-09679-7

Chapter One

Fundamental ideas

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).

(iv) When K contains the finite field [F.sub.q], the mapping

[PHI] : [F.sub.q] [right arrow] [F.sub.q]

x [??] [x.sup.q],

is the Frobenius automorphism. The n-th Frobenius automorphism of K is the map [[PHI].sup.n] that takes x [element of] K to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [element of] K. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] consists of all elements in K which are fixed by [[PHI].sup.n].

The Frobenius collineation associated to the Frobenius automorphism is the collineation of PG(n, K) with [sigma] = [PHI]; that is,

x [??] x', [sup.t]x' = A [sup.t][PHI](x),

with [PHI]F(x) = ([x.sup.q.sub.0], [x.sup.q.sub.1], ..., [x.sup.q.sub.n]), for some non-singular matrix A.

Remark 1.3 (i) When K = [F.sub.q], it is customary to replace K by q in the notation for all the spaces and groups; so V(n, q) means V(n, K), and similarly for AG(r, q), PG(r, q), GL(r, q), [GAMMa]L(r, q), AGL(r, q), PGL(r, q), and P[GAMMA]L(r, q).

(ii) When K is algebraically closed and has characteristic p > 0, then K contains the finite field [F.sub.q] for every power of q of p.

1.2 POLYNOMIALS

Definition 1.4 (i) A polynomial f in the ring K][X.sub.1], [X.sub.2], ..., [X.sub.n]]of polynomials in the indeterminates [X.sub.1], [X.sub.2], ..., [X.sub.n] is reducible if there exist non-constant [f.sub.1], [f.sub.2] in K][X.sub.1], [X.sub.2], ..., [X.sub.n]] with f = [f.sub.1][f.sub.2]; otherwise, f is irreducible.

(ii) The degree of a monomial [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [r.sub.1] + [r.sub.2] + ... + [r.sub.n].

(iii) A polynomial is homogeneous if all its terms have the same degree.

(iv) The degree of a polynomial f is the largest degree of all its terms; write deg f.

1.3 AFFINE PLANE CURVES

In the first instance, a curve is associated both to a set of points and to a polynomial. Let K be an algebraically closed field, and let F [element of] K[X, Y]. Then an affine curve is viewed as a set of points.

Definition 1.5 (i) The plane affine curve

F = [v.sub.a](F) = {P = (x, y) [element of] AG(2,K)| F(x, y) = 0}.

(ii) The degree of F, written deg F, is deg F.

Any affine transformation sends an affine curve to another having the same degree. Therefore deg F of an affine curve F is an affine invariant.

Definition 1.6 (i) A component of the affine curve F = [v.sub.a](F)is an affine curve G = [v.sub.a](G) such that G divides F.

(ii) The affine curve F = [v.sub.a](F) is irreducible when it has no proper component, that is, when F is irreducible.

Components are covariant, that is, the diagram below is commutative for any affine transformation T of AG(2, K).

Any line containing at least n + 1 points from an affine curve F of degree n is a component of F. To show this, it may be assumed by covariance that l = [v.sub.a](Y). Let F = [v.sub.a](F(X, Y)). Then | l [intersection] [v.sub.a](F)| [greater than or equal to] n + 1 implies that F(X, 0) has more than n roots. Therefore F(X, 0) = 0, and hence X divides F(X, Y).

Let F = [v.sub.a](F) be an affine curve with deg F = d, and let l = bX - aY + c be a line containing the point [P.sub.0] = ([x.sub.0], [y.sub.0])on F. Then, for any point P = (x, y) [element of] l,

[[b.sub.x]] - ay = [bx.sub.0] - [ay.sub.0],

b(x - [x.sub.0]) = a(y - [y.sub.0]) = abt,

x = [x.sub.0] + at, y = [y.sub.0] + bt

for some t [element of] K. Then

F (x, y)= F ([x.sub.0] + at, [y.sub.0] + bt)= G(t)= [G.sub.0] + [G.sub.1] t + [G.sub.2][t.sup.2] + + [G.sub.d][t.sup.d] = [G.sub.m][t.sub.m] + + [G.sub.d][t.sup.d], (1.1)

with [G.sub.m] [not equal to] 0, [G.sub.d] [not equal to] 0.

Lemma 1.7 The two irreducible curves [F.sub.1] = [v.sub.a]([F.sub.1]) and [F.sub.2] = [v.sub.a]([F.sub.2]) are the same if and only if [F.sub.2] = [lambda][F.sub.1] for some [lambda] [element of] K\{0}.

Proof. This is a consequence of Theorem 2.10.

Definition 1.8 If F [element of] K[X, Y] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with each [F.sub.i] irreducible, then F = [v.sub.a](F) has components [F.sub.i] [v.sub.a]([F.sub.i]) with multiplicity [n.sub.i] for i = 1, ..., s.

The multiplicity of a component is an affine invariant.

Definition 1.9 Let l be a line which is not a component of F.

(i) The integer m of (1.1) is the intersection number of l and F at [P.sub.0]: write m = I([P.sub.0], l [intersection] F); (ii) if m = 1 for some line l through [P.sub.0], then [P.sub.0] is a simple or non-singular point of F;

(iii) if m [greater than or equal to] 2 for all lines l through [P.sub.0], then [P.sub.0] is a singular or multiple point of F;

(iv) if [m.sub.0] = min{m | l a line through [P.sub.0]}, then [m.sub.0] is the multiplicity of [P.sub.0] on F, or [P.sub.0] is an [m.sub.0]-fold point of F, and write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(v) if m > [m.sub.0] for a line l, then l is a tangent to F at [P.sub.0].

The intersection number and the multiplicity of a point are affine invariants.

Definition 1.10 If [m.sub.P](F) = 2, then P is a double point of F. A double point P with two distinct tangents to F at P is a node, and with only one tangent to F at P is a cusp. If [m.sub.P](F) = 3, then P is a triple point of F.

Remark 1.11 Let M be a subfield of K and suppose that F is defined over M, that is, F = v(f(X, Y))with f(X, Y) [element of] M[X, Y]. If P is a double point with two distinct tangents, neither of them defined over M, then P is an isolated double point over M.

Lemma 1.12 If [P.sub.0] is a simple point of F, then, in (1.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary 1.13 The tangent to F at a simple point P = (x, y) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note the meaning of this corollary: the line [l.sub.P] has intersection multiplicity at least 2 with F at P.

Definition 1.14 A non-singular point P of F is a point of inflexion of F if

I(P, [l.sub.P] [intersectioin] F) [greater than or equal to] 3.

Here, P is also called an inflexion or, in some sources, a flex; the tangent [l.sub.P] at P is the inflexional tangent. Tangents and inflexional tangents are covariant.

(Continues...)

Excerpted from Algebraic Curves over a Finite Fieldby J.W.P. Hirschfeld G. Korchmros F. Torres Copyright İ 2008 by Princeton University Press. Excerpted by permission.
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Edition details

• Publisher: Princeton University Press
• Publication date: 2008
• Language: English
• ISBN 10: 0691096791
• ISBN 13: 9780691096797
• Binding: Hardcover
• Number of pages: 744