Personally i was attracted by the rich examples explained in this book. Finite fields and their applications 5, 393408 1999. Coding theory and algebraic curves over finite fields. To complete the proof of existence, we have to check that jfpnj pn. In mathematics, a finite field or galois field sonamed in honor of evariste galois is a field that contains a finite number of elements. Xing, towers of global function fields with asymptotically many rational places and an improve ment on the gilbertvarshamov bound, math. Let lbe a vector space of rational functions with prescribed behavior of their poles and zeros. Algebraic curves over a finite field princeton university press. The principal concept is that of using algebraic curves over finite fields to construct errorcorrecting codes. University of groningen algebraic curves over finite. The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive. Algebraic curves over a finite field are the basic structure in the theory of algebraicgeometry codes, which combines algebraic geometry and errorcorrecting codes. It is clear that u is a subset of f q since f qn is a subset of f p. A variety of dimension one is called an algebraic curve.
F torres this title provides a selfcontained introduction to the theory of algebraic curves over a finite field, whose origins can be traced back to the works of gauss and galois on algebraic equations in. In 11, a new bound for the number of points on an algebraic curve over a nite. From string theory to elliptic curves over finite field, f. Thenweusethe connectionbetween the theory of linear codes and algebraic curves over finite fields goppacodes10. In the sequel, an algebraic curve means an irreducible algebraic curve over an algebraically closed field. Cryptology, designs and finite groups cdfg 2009 groups and algebraic curves over a. In this paper, we prove that a smooth projective curve over a finite field can be recovered from lfunctions associated to the hilbert class field of the curve and its constant field extensions.
On the other hand, most books with a modern approach demand considerable background in algebra and topology, often the equivalent of a year or more of graduate study. From string theory to elliptic curves over finite field, fp a senior project submitted to the division of science, mathematics, and computing of bard college by linh thi dieu pham annandaleonhudson, new york may, 2014. The riemann hypothesis for curves over finite fields. F torres this title provides a selfcontained introduction to the theory of algebraic curves over a finite field, whose origins can be traced back to the works of. We will use the language of smooth projective curves and compact riemann surfaces interchangeably. Request pdf on algebraic curves over a finite field with many rational points in 12, a new upper bound for the number of doublestruck f signqrational points on an absolutely irreducible. Algebraic complexities and algebraic curves over finite.
Algebraic codes on lines, planes, and curves richard e. It is clear that the algebraic closure f q of f q is the same as f p. Request pdf algebraic curves over a finite field this book provides an accessible and selfcontained introduction to the theory of algebraic curves over a finite field, a subject that has. Dieudonnl received june 18, 1984 let x c, x cz be the product of two nonsingular projective curves defined over. Rational points and places over a finite field 277 8. Rational points on curves over finite fields by harald. Algebraic curves over a finite field request pdf researchgate.
Pdf on the number of rational points on an algebraic curve over a. The arithmetic of the product of two algebraic curves over a. Algebraic curves over finite fields cambridge tracts in. After all, you are just listing a finite set of points. Pdf a new bound for the number of rational points on an algebraic curve over a finite field is obtained in theorem 1. Up to birational equivalence, the irreducible curves over a field f are categorically equivalent to algebraic function fields in one variable over f. The deepest results on the number of points of an algebraic curve over fq, such as the hasseweil theorem, the serre bound and the st ohrvoloch theorem, are formulated for irreducible, nonsingular algebraic curves. Algebraic complexities and algebraic curves over finite fields. The paper on the number of curves of genus 2 over a finite field by gabriel cardona determines the actual number of isomorphism classes of genus two curves over a finite field of odd characteristic. We will assume all curves are over the complex numbers.
Electrical engineers as well as mathematics students will find the material of interest. Presents an introduction to the theory of algebraic curves over a finite field, a subject that has applications in areas such as finite geometry, number theory, errorcorrecting codes, and this book emphasizes the algebraic geometry rather than the function field approach to algebraic curves. Such an algebraic function field is a field extension k of f that contains an element x which is transcendental over f, and such that k is a finite algebraic extension of f x, which is the. This book provides an accessible and selfcontained 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, errorcorrecting codes, and cryptology. It is a prime eld in the sense that there are no proper sub elds of f p. Combinatorial structures in finite projective spaces see also 05bxx keywords algebraic curve finite field arc. Algebraic curves over a finite field are the basic structure in the theory of algebraic geometry codes, which combines algebraic geometry and errorcorrecting codes. Algebraic curves over a finite field is a rich, examplefilled, comprehensive introduction to the subject. Request pdf on algebraic curves over a finite field with many rational points in 12, a new upper bound for the number of doublestruck f signq.
We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with. Algebraic curves with many points over finite fields fernando torres as long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. Request pdf algebraic curves over a finite field this book provides an accessible and selfcontained introduction to the theory of algebraic curves over a. Algebraic curves, finite fields, rational points, genus, linear codes, asymp totics, tower of curves. The simplest and clearest concept is that of a plane affine algebraic curve. If you consider the points on the curve over an extension field, then what are they. You can fit many curves through this set of points, so you havent really defined a. As an easytoread introductory book that presents the general theory of algebraic curves over finite fields, it fills a large gap in the literature. Masaaki homma, kanagawa university this book is wellwritten and i greatly enjoyed reading it. The most recent developments are presented including the theory of codes on curves, without the use of detailed mathematics, substituting the intense theory of algebraic geometry with fourier transform where possible. Masters thesis algebraic curves over finite fields.
We know that in algebraic geometry see basics of algebraic geometry, the points of a curve or more general varieties correspond to maximal ideals of a ring of functions on the curve. This is a survey on recent results on counting of curves over finite fields. When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to comprise all nonsingular cubic curves. The arithmetic of the product of two algebraic curves over. Ouralgorithmsofpolynomialmultiplicationcanbeinterpretedas interpolation methodsonalgebraiccurves. The set of algebraic curves defined over a global field. Chudnovsky department of mathematics, columbia university, new york, new york 100. On algebraic curves over a finite field with many rational.
Here, only a brief exposition of the main construction, due to goppa, and. The most common examples of finite fields are given by the integers mod p when. But when these sciences joined company they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. Algebraic complexities and algebraic curves over finite fields d. Since goppas construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here, only a brief exposition of the main construction, due to goppa, and a few illustrative examples are presented in section. Pdf recovering algebraic curves from lfunctions of. This book is a selfcontained guide to the theory of algebraic curves over a finite field, one that leads readers to various recent results in this and related areas. An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g. As a consequence, we give a new proof of the theorem of mochizuki and tamagawa that an isomorphism between the fundamental groups of two such curves comes from an. University of groningen algebraic curves over finite fields. Algebraic curves with many points over the binary field. Here we explain the main tools for finding rational points on a curve over a finite field and provide the necessary background on ring and field theory. Recently, the authors discovered another important application of such curves, namely to the construction of.
Indeed, by construction, the elements of fpn are the set of roots to the polynomial ft tpn t. Algebraic curves over a finite field princeton series in. The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. Algebraic curves over finite fields with many rational points have received a lot of attention in recent years. Ihara, some remarks on the number of rational points of algebraic curves over finite fields, j. You are advised to consult the publishers version publishers pdf if you. Defined over the finite field means that the polynomial equation that defines the curve has coefficients which are elements of the finite field.
The theory of algebraic curves over finite fields, their zeta and lfunctions, and, for the first time, the theory of algebraic geometric goppa codes on algebraic curves are developed in this text. For infinite fields minimal complexities are known winograd, s. Our new algorithms can be interpreted as interpolation methods on algebraic curves. Some people studied complex analysis of riemann surfaces, and others studied polynomials in two variables. Curves over finite and local fields see also 14h25 secondary.
Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point o. The number of points on an algebraic curve over a nite eld. On curves over finite fields by arnaldo garcia emis. Journal of algebra 98, 102142 1986 the arithmetic of the product of two algebraic curves over a finite field noriko yur department of mathematics, university of toronto, toronto, ontario m5s iai, canada communicated by j.
We present a survey of this subject covering both the case of fixed genus and the asymptotic theory. The arithmetic of the product of two algebraic curves over a finite field noriko yur department of mathematics, university of toronto, toronto, ontario m5s iai, canada communicated by j. Wepresentall the relevant informationfromthetheoryofalgebraiccurvesoverfinite fields. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
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