We will often view fas the primary object of interest, and in this case refer to eas an extension eld or simply extension of f. Felipe voloch preprints university of texas at austin. Associated with some systems of unramified coverings of algebraic curves over finite fields there are spaces analogous to the universal covering transformation spaces. Rational points on cubic, quartic and sextic curves over finite fields 3 for a curve c over fq and n. Introduction padic numbers satohs algorithm introduction padic numbers satohs algorithm fre vercauteren. This chapter describes the specialised facilities for elliptic curves defined over finite fields. In this chapter we study elliptic curves defined over a finite field.
Elliptic curves over finite fields and the computation of square roots mod p by rene schoof abstract. Elliptic curves over finite fields and the computation of. Rational points on curves over finite fields theory and applications. Counting points on curves over finite fields springerlink. Nnc is the number of rational points of a curve c over fqn. Elliptic curves over finite fields indian institute of. Enumerating curves of genus 2 over finite fields university of.
We show that there exists a subgroup r of finite index. Finite descent obstruction for curves over function fields, j. You may be more familiar with vector spaces over cand r. Enumerating curves of genus 2 over finite fields presentedby rosek. Group structures of elliptic curves over finite fields vorrapan chandee, chantal david, dimitris koukoulopoulos, and ethan smith abstract. Special semester on applications of algebra and number theory. Extension fields throughout these notes, the letters f, e, kdenote elds. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of lowdiscrepancy sequences. Curves over finite fields 5 we will see improvements on this bound in later sections of this article.
In computational perspectives on number theory chicago, il, 1995, volume 7 of amsip stud. In practice it makes z a rational function of t, something that is interesting even in the case of v an elliptic curve over finite field. We prove the following result which was conjectured by stichtenoth and xing. Weibel received 5 may 1986 let x be a smooth projective curve over a finite field.
Alina bucur, ucsd for background material, and more details about the projects, see the notes. Cambridge core geometry and topology algebraic curves over finite fields by carlos moreno. The number of points on a curve defined over a finite field is bounded as a function of its genus. The historical development of errorcorrecting codes starts with hamming codes, and looks at other linear codes such as reedsolomon, bosechaudhurihocquengham, and goppa codes. On the number of rational points on an algebraic curve over a finite field. Algebraic complexities and algebraic curves over finite fields d.
Lectures on zeta functions over finite fields daqing wan department of mathematics, university of california, irvine, ca926973875 email. This work studies the arithmetic of polynomial rings over finite fields, af t, and its relation to elementary number theory, which is concerned with the arithmetic properties of the ring of integers, z, and its field of fractions, the rational numbers, q. You are advised to consult the publishers version publishers pdf if you. 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. For d a divisor of a curve c, nd qdegd is the norm of d. As a group, the jacobian variety of a curve is isomorphic to the quotient of the group of divisors of degree zero by the subgroup of principal divisors, i. Why is it that it cannot be the product of more than two cyclic groups. Algebraic curves over nite elds in this chapter, we will recall some basic knowledge of nite elds, schemes theory and algebraic curves. Algebraic curves over a finite field is a rich, examplefilled, comprehensive introduction to the subject. Furthermore it is known that every finite group occurs in this way, since, for any ground field k and any finite group g, there exists an algebraic curve x defined over k such that autx. In 2005, this was embedded in a larger project of determining n. In this section, we define algebraic sets, which are the most basic objects of algebraic geometry.
Introduction to finite fields for further reading on this beautiful subject, see e. Algebraic curves over finite fields with many rational points have received a lot of attention in recent years. Most of the machinery has been constructed with elliptic curve cryptography in mind. Coombes department of mathematics, university of michigan, ann arbor, mi 48109, u.
There are two types of belyis theorem for curves defined over finite fields of characteristic p, namely the wild and the tame pbelyi theorems. This theory can be applied to the reduction mod unk of the shimura curves. This can be done over any eld over which there is a rational point. Felipe voloch old preprints university of texas at austin. The most important arithmetic quantity associated with such a curve is its number of rational points.
Elliptic and modular curves over finite fields and related computational issues. This book collects the results of the workshops applications of algebraic curves and applications of finite fields of the ricam special semester 20. For infinite fields minimal complexities are known winograd, s. 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. These workshops brought together the most prominent researchers in the area of finite fields. The genus of curves over finite fields with many rational. Decomposing jacobians of curves over finite fields in the. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. Open questions on jacobians of curves over finite fields. Chudnovsky department of mathematics, columbia university, new york, new york 100. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.
Rational points on curves and surfaces over finite fields. Equations singular points the discriminant elliptic curves f 2 elliptic curves f 3 the sum of points examples structure of ef 2 structure of ef 3 further examples notations fields of characteristics 0 1 q is the. Syllabus elliptic curves mathematics mit opencourseware. For any projective embedding of a nonsingular irreducible complete algebraic curve defined over a finite field, we obtain an upper bound for the number of its rational points. Those involve different finite fields for example the whole family of fields zpz as p runs over all prime numbers. Algebraic complexities and algebraic curves over finite. Rational point counts for del pezzo surfaces over finite fields and coding theory abstract the goal of this thesis is to apply an approach due to elkies to study the distribution of rational point counts for certain families of curves and surfaces over nite elds.
Masters thesis algebraic curves over finite fields. This holds for fields that are not algebraically closed, provided one considers divisors and functions defined over that field. Details concerning their construction, arithmetic and basic properties may be found in chapter elliptic curves. In this introductory article, we survey what is known about the maximum number of points on a curve of genus g defined over. Counting points on varieties over finite fields of small characteristic. Algebraic curves over finite fields by carlos moreno. Computing zeta functions of curves over finite fields. Here, the authors relate an important application of such curves, namely, to the construction of lowdiscrepancy sequences, needed for numerical methods in diverse areas.
Linearizing torsion classes in the picard group of algebraic curves over finite fields. Curves over finite fields whose cardinality is a square attaining the. Since goppas construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. This is a survey on recent results on counting of curves over finite fields. This thesis surveys the issue of finding rational points on algebraic curves over finite fields. We study the distribution of algebraic points on curves in abelian varieties over finite fields.
Published tables of irreducible polynomials over finite fields are insufficient to factor xn 1 for even modest values of n. Elliptic curves over f q introduction history length of ellipses why elliptic curves. We study whether the set of rational points of a curve over a finite field generates the set of rational points of its jacobian. Construction of codes not to mention efficient encodingdecoding algorithms requires that we develop basic facts about finite fields and linear algebra over them. Knapp, advanced algebra, digital second edition east setauket, ny. Here, only a brief exposition of the main construction, due to goppa, and. We study hyperbolic curves and their jacobians over finite fields in the context of anabelian geometry.
We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. Nq denote an algebraic closure of q, and qk the subfield of nq of order q. The correspondence between the curves over number fields and dessins. But the results were stated for curves over complex numbers. Abstractwe address the problem of computing in the group of. Errorcorrecting codes, finite fields, algebraic curves.
Rational points on algebraic curves over finite fields is a key topic for algebraic geometers and coding theorists. Constructing isogenies between elliptic curves over finite fields volume 2 steven d. In number theory, the local zeta function zv, s sometimes called the congruent zeta function is defined as. For any prime pand any nonzero integer m, there exists a. We start by proving a theorem of hasse which says that if k is a field with q elements, and e k is an elliptic curve, then e k contains approximately q points, with an. Is it true over finite fields too, and are there caveats. Some fundamental groups in the arithmetic of algebraic. Algebraic curves over finite fields 1115 november 20. The constants in the bound are related to the weierstrass order.
Thenweusethe connectionbetween the theory of linear codes and algebraic curves over finite fields goppacodes10. Genus2 hyperelliptic curves over finite fields byencinas, menezes, andmasqu. On the ktheory of curves over finite fields sciencedirect. Rational points on curves over finite fields by harald. In fact the full panoply of linear algebrathe concepts of basis, dimension, linear transformation. University of groningen algebraic curves over finite. We present a survey of this subject covering both the case of fixed genus and the asymptotic theory. Could anyone explain why the abelian group of points on an elliptic curve over a finite field is isomorphic to at most two cyclic groups. The theory of algebraic curves over finite fields has its origins in number theory. Lets ignore the deeper reason for this and simply say that if the characteristic were two or.
Introduction these are the notes from the summer school in gottingen sponsored by nato advanced study institute on higherdimensional geometry over finite fields that took place in 2007. On maximal curves over finite fields of small order. Berlekamp, algebraic coding theory, aegean press, 1984, r. In essence coding theory is a game where one tries to. A vector space of polynomials over a xed nite eld f q gives rise to. This is a survey on recent results on counting of curves over. It is a prime eld in the sense that there are no proper sub elds of f p. Introduction to finite fields this example illustrates that the group structure i. Constructing isogenies between elliptic curves over finite. Journal of pure and applied algebra 51 1988 7987 79 northholland on the ktheory of curves over finite fields kevin r.
In this paper we present a deterministic algorithm to compute the number of fpoints of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation. Knapp, 2016, 2016 rationally connected varieties over finite fields kollar, janos and szabo, endre, duke mathematical journal, 2003. However, many applications of curves were found in different areas such as coding theory, sphere packings and lattices, sequence design, quasimonte carlo methods, and cryptography. Elliptic curves over finite fields department of mathematics. The chordtangent method does give rise to a group law if a point is xed as the zero element. Finite fields as vector spaces s uppose that fis a nite eld of characteristic p, with prime sub eld p f p.
As an easytoread introductory book that presents the general theory of algebraic curves over finite fields, it fills a large gap in the literature. Find materials for this course in the pages linked along the left. Onelevel density for cyclic trigonal curves the rst project concerns the computation of the average number of points of f qn, the one. Counting points on elliptic curves over finite fields and beyond. It is wellknown that if e is an elliptic curve over the nite eld f p, then ef p zmz zmkz for some positive integers m. Before we continue, we make the standing assumption that the characteristic of fq is not two or three, i. Algebraic complexities and algebraic curves over finite fields.
811 1473 81 1016 1250 662 174 1268 961 1219 1206 113 1100 280 1207 475 62 769 562 1512 539 477 634 107 686 1214 3 556 1443