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Algebraic Curves over a Finite Field

Algebraic Curves over a Finite Field

J.W.P. Hirschfeld
G. Korchmáros
F. Torres
Copyright Date: 2008
Edition: STU - Student edition
Pages: 720
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  • Book Info
    Algebraic Curves over a Finite Field
    Book Description:

    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.

    eISBN: 978-1-4008-4741-9
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. Preface
    (pp. xi-xx)

    • Chapter One Fundamental ideas
      (pp. 3-20)

      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-singuh points have a point in ccmmon.

      Over the real numbers,${\mathbf {R}}$, consider the parabola${\mathcal {F}}$given byF=YX2; its points form, as in Figure 1.1, the set\[\{(t,{{t}^{2}})\, |\, t\in \mathbf {R}\}.\]

      However, there are two other types of point associated with${\mathcal {F}}$, namely, (a) those at infinity and (b) those wih coordinates in C, the algebraic closure of${\mathbf {R}}$....

    • Chapter Two Elimination theory
      (pp. 21-36)

      The intersection number$I(P,\ell \cap {\mathcal {F}})$of an algebraic plane curve${\mathcal {F}}$and a line ℓ at a pointPwas used in Chapter 1 for the study of local projective invariants, such as singularities and inflexions. In this and subsequent chapters, a natural extension of$I(P,\ell \cap {\mathcal {F}})$to the intersection number$I(P,{\mathcal {G}} \cap {\mathcal {F}})$of${\mathcal {F}}$and another algebraic plane curve${\mathcal {G}}$atPis investigated. The main outcome, Bézout’s Theorem 3.14, is an equation linking the degrees of${\mathcal {F}}$and${\mathcal {G}}$to the intersection numbers at their common points.

      Consider${\mathcal {F}}={\mathbf {v}}(f(X,Y))$,$\ell ={\mathbf {v}}(uY - X)$andP= (0, 0), such that ℓ is not...

    • Chapter Three Singular points and intersections
      (pp. 37-62)

      The aim of Section 3.1 is to define the intersection number$$I(P,{\mathcal {F}} \cap {\mathcal {G}})$$of two plane curves at a pointPby means of purely geometric postulates. The main results, Theorems 3.8 and 3.9, ensure that such a number not only exists but is also uniquely determined. Also, forP= (0, 0), this definition coincides with that arising from elimination theory as described at the beginning of Chapter 2.

      In Section 3.2, the concept of intersection number is extended to include projective curves${\mathcal {F}}$and${\mathcal {G}}$and pointsPat infinity. With this, Bézout’s Theorem 3.14 can be formulated and...

    • Chapter Four Branches and parametrisation
      (pp. 63-109)

      From now on, only projective curves are considered, and the adjective ‘projective’ is omitted.

      Concepts such as the degree of a curve, the multiplicity of a point on a curve, and the number of singular points of a curve are all projective invariants.

      The resolution of singularities in Chapter 3 required a broader class of transformations, namely birational transformations: projective transformations were not sufficient. Any projective invariant is also a birational invariant. So the question arises of the existence of birational invariants; that is, non-negative integers that are constant under a birational transformation that does not necessarily fix a projective...

    • Chapter Five The function field of a curve
      (pp. 110-160)

      A generalisation of the idea of a branch is a generic point of an irreducible plane curve${\mathcal {F}}={\mathbf {v}}(F)$. The coordinates of a generic point are in some proper extension ofK, but still are assumed to satisfy the polynomialF. As in Section 4.2, branch representations are points in this sense.

      The coordinates of a generic point of${\mathcal {F}}$together withKgenerate a field Σ of transcendence degree 1 overK. Distinct generic points of${\mathcal {F}}$give rise to the same field Σ, the function field$K({\mathcal {F}})$associated to${\mathcal {F}}$, determined up to aK-isomorphism. It may...

    • Chapter Six Linear series and the Riemann–Roch Theorem
      (pp. 161-198)

      In the investigation of the birational invariants of an irreducible plane curve, a central role is played by divisors, which are finite formal sums of places, and by linear series, which are certain sets of effective divisors. This chapter develops the theory of linear series; this includes the Riemann–Roch Theorem 6.61, the Weierstrass Gap Theorem 6.89 and the study of divisor class groups, which are also called Picard groups.

      As in the previous chapter, the field Σ =K(x, y) is associated to an irreducible algebraic curve${\mathcal {F}}$equipped with a generic pointP= (x, y). From an...

    • Chapter Seven Algebraic curves in higher-dimensional spaces
      (pp. 199-274)

      Algebraic curves in higher-dimensional spaces are usually introduced in the more general context of arbitrary algebraic varieties. However, an equivalentad hocdefinition which fits in with the plan of this book is also possible. Essentially, an irreducible algebraic curve of PG(n, K) is a set of points which can be put in a birational correspondence with the points of an irreducible algebraic plane curve. The idea of deriving the concept of an algebraic curve in a higher-dimensional space directly from that of an algebraic plane curve has at least two advantages. On the one hand, it allows the development...


    • Chapter Eight Rational points and places over a finite field
      (pp. 277-331)

      In this chapter,$K=\overline{{{\mathbf F}}}_{q}$, the algebraic closure of the finite field${{\mathbf {F}}_{q}}$of orderq. For anyr, the space PG(r,K) contains the finite projective spaces PG(r,qi) withi≥ 1.

      If Γ is an algebraic curve embedded in PG(r,K), the set of points of Γ lying in PG(r,q) is a natural geometric object to investigate. To do this in a birational spirit, the concept of an${{\mathbf {F}}_{q}}$-rational place, that is, a place defined over${{\mathbf {F}}_{q}}$, is required. As in Section 5.3, the idea is to derive this concept from that of an...

    • Chapter Nine Zeta functions and curves with many rational points
      (pp. 332-392)

      In this chapter as in the last,$K=\overline{{{\mathbf F}}}_{q}$, the algebraic closure of${{\mathbf {F}}_{q}}$. Let Γ be an irreducible algebraic curve of PG(r,K) defined over${{\mathbf {F}}_{q}}$and equipped with the places of its function field Σ. By definition, as in Section 7.1, Γ arises from an irreducible plane curve${\mathcal {F}}$defined over${{\mathbf {F}}_{q}}$. If${\mathcal {F}} =\mathbf {v}(f(X,Y))$withf(X,Y) in${{\mathbf {F}}_{q}}[X,Y]$, then Σ =K(x,y) withf(x,y) = 0.

      The aim here is to investigate birational properties of Γ using its${{\mathbf {F}}_{q}}$-rational points and, in particular, the number of its${{\mathbf {F}}_{q}}$-rational places. For...


    • Chapter Ten Maximal and optimal curves
      (pp. 395-457)

      In the study of curves with many${{\mathbf {F}}_{q}}$-rational points the main problem is to determineNq(g), the largest number of${{\mathbf {F}}_{q}}$-rational points that an irreducible${{\mathbf {F}}_{q}}$-rational curve of genusgcan have. The asymptotic behaviour ofNq(g) with respect togtogether with the Drinfeld–Vlăduţ Bound (9.38) are discussed in Section 9.3, where some examples with${{N}_{q}}(g)\approx (\sqrt{q}-1)g$for a fixed squareqand forglarge enough are also constructed. In general,Nq(g) is known for only a few pairs (q,g).

      The aim of this chapter is threefold:

      (1) to describe the theory of${{\mathbf {F}}_{q}}$...

    • Chapter Eleven Automorphisms of an algebraic curve
      (pp. 458-545)

      The concept of a symmetry, or automorphism in modern terminology, plays a prominent role in all branches of geometry. The saying,the larger its automorphism group the richer its geometry, is especially appropriate for algebraic curves. For instance, if there exists an automorphism of an irreducible curve Γ, then another irreducible curve, the quotient curve, can be constructed, and it inherits important properties of Γ. Also, certain families of curves have characterisations obtained from the structure and action of their automorphism groups.

      As in several concepts introduced previously, an automorphism of an algebraic curve is a birational invariant. In the...

    • Chapter Twelve Some families of algebraic curves
      (pp. 546-589)

      The aim of this chapter is to provide families of algebraic curves in positive characteristic with properties that a complex algebraic curve cannot have. As mentioned in the earlier chapters, these curves illustrate concepts and results unknown in the classical literature on curves.

      In this section, the curve with\[{\mathcal {C}}=\mathbf{v}(A(Y)-B(X)),\caption {(12.1)}\]is investigated in detail under the following conditions:

      (I)$\deg \mathcal {C}\ge 4$and gcd(p, degB(X)) = 1;

      (II)$A(Y)={{a}_{n}}{{Y}^{{{p}^{n}}}}+{{a}_{n-1}}{{Y}^{{{p}^{n-1}}}}+\cdots +{{a}_{0}}Y,\quad {{a}_{j}}\in K,\,{{a}_{0}},\,{{a}_{n}}\ne 0$;

      (III)$B(X)={{b}_{m}}{{X}^{m}}+{{b}_{m-1}}{{X}^{m-1}}+\cdots +{{b}_{1}}X+{{b}_{0}},\quad {{b}_{j}}\in K,\ {{b}_{m}}\ne 0$;

      (IV)m≢ 0 (modp);

      (V)n≥ 1,m≤ 2.

      Note that (II) occurs if and only ifA(Y+a) =A(Y) +A(a)...

    • Chapter Thirteen Applications: codes and arcs
      (pp. 590-626)

      Algebraic curves over a finite field are the basic structure in the theory of algebraic-geometry codes, which combines algebraic geometry and error-correcting codes.

      Here, only a brief exposition of the main construction, due to Goppa, and a few illustrative examples are presented in Section 13.1.

      Coding theory is also connected with algebraic curves via finite geometry, since complete arcs in PG(r,q) are the geometric counterpart of maximum distance separable (MDS) codes, which are linear codes correcting the greatest number of errors with respect to their length and dimension. In the other sections, an account of this topic, especially on...

  7. Appendix A. Background on field theory and group theory
    (pp. 627-649)
  8. Appendix B. Notation
    (pp. 650-654)
  9. Bibliography
    (pp. 655-688)
  10. Index
    (pp. 689-696)