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Algebraic Curves over a Finite Field (Princeton Series in Applied Mathematics), by J. W.P. Hirschfeld, G. Korchmáros, F. Torres
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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.
- Sales Rank: #660396 in Books
- Brand: Brand: Princeton University Press
- Published on: 2008-03-23
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.50" w x 6.14" l, 3.51 pounds
- Binding: Hardcover
- 744 pages
- Used Book in Good Condition
Review
"This book is well-written and I greatly enjoyed reading it. The wealth of information and examples in this book give the reader a firm foundation and develop an intuition for the subject. The authors have used it as a textbook for a two-year course, and it would be a fine introduction to any advanced undergraduate or graduate student wanting to learn this subject."--Thomas Hagedorn, MAA Reviews
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
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.
Most helpful customer reviews
3 of 4 people found the following review helpful.
A good introduction
By Dr. Lee D. Carlson
The content of this book is not only intrinsically interesting but also of great importance to applications such as coding theory and cryptography. The authors' emphasis of course is on algebraic curves over fields of non-zero characteristic, but in the initial chapters they formulate the results as much as possible to be independent of characteristic. The first chapter for example is a review of the elementary algebraic geometry of affine and projective curves over any characteristic, but the authors also give examples of some of the peculiarities of algebraic curves in non-zero characteristic, such as the existence of "strange curves." A strange curve is one that has a "nucleus", i.e. a point that is included in all tangents to the curve at all non-singular points. The projective line is the only non-singular strange curve in characteristic zero, and so visual representations of strange curves will be elusive. Readers will have to rely purely on algebra for their understanding.
In these early chapters readers will also get their first taste of intersection theory, a subject that has taken on particular importance lately due to its connections with theoretical physics. Characterizing when two algebraic plane curves intersect entails eliminating one of the variables, and so the authors give a fairly detailed discussion of elimination theory and prove the classical theorem of Bezout. This is followed by definitions of rational and birational transformations, and then to a proof of the resolution of singularities theorem. This proof is done independent of characteristic, but the reader can see just where peculiarities can arise in the case of non-zero characteristic. These peculiarities arise from the existence of strange curves, but the authors show explicitly how to transform these curves so that the proof will hold. It should also be noted that the resolution of singularities theorem proved in this book is only for plane algebraic curves, and not the general one proved by Hironaka in characteristic zero and other mathematicians for certain non-zero values of the characteristic.
The authors do not motivate very well why they are going to analyze branches of curves using formal power series rather than just polynomials. A bit of historical background would have been good, instead of merely referring the reader to a book that is out-of-print. Readers familiar with the theory of ordinary singular differential equations will recognize some of the terminology, and this knowledge will in fact allow a better appreciation of some of the concepts in the discussion on branch representations. In the case of differential equations for example it is frequently useful to use power series representations for the coefficients of these equations, and to construct generalizations of Newton diagrams called Puiseux diagrams.
The reader will have to be patient and finally see their application to the study of algebraic plane curves after several pages of discussion on branch representations, which arise from the field of rational functions of formal power series. The goal here is to define a formal parameterization of the (projective) algebraic curve involving the ring of formal power series in an indeterminate t, or loosely speaking a "parametric" representation in the "parameter" t. As a trivial example all readers should be familiar with is the unit circle in the plane, which can be parameterized by the ordinary trigonometric functions sin(t) and cos(t) both of which have power series representations in the parameter t. The interesting cases arise when the curve has singularities, and this is the reason for bringing in the field of rational functions. In the authors' view, a branch representation is point in the projective plane (over the field of rational functions of formal power series in a single indeterminate) that is contained in the ordinary projective plane. The authors show how to obtain the simplest, i.e. the most "primitive" branch representation, and indicate when two branch representations are equivalent (change of parameters). They also show how to transform the primitive branch representations to "imprimitive" ones, using a monomorphism of the field of rational functions that is not an automorphism of the underlying field. The order of this monomorphism is the `ramification index' of the imprimitive branch representation. Using the notion of equivalence, they define a `branch' to be an equivalence class of primitive branch representations.
The utility of having the concept of a branch for the study of plane algebraic curves comes from defining a branch of such a curve to be a branch whose representations are the zeros of the defining polynomial of the curve. The notion that a branch is a "piece" of the curve that is smooth and continuous and the largest of such pieces is brought out in the notion of a `center' of a branch. Speaking somewhat loosely, this is the point where the parameter t is taken to be zero, and the authors show that this is a point of the curve. They then show that there is a unique branch of the curve that is centered at a simple point of the curve. Using this result, and their proof that the intersection multiplicity of an irreducible plane curve F with a singular point P and a curve G containing F as a component can be written as the sum of the intersection multiplicities of all branches of F centered at P, the reader can easily appreciate the idea that branches of a curve are the pieces as characterized above.
Taking the formal sum of branches leads to the notion of a `divisor', which is very important in modern algebraic geometry and is the source of many brilliant developments in the field. Using it, one is also able to connect geometric results with purely algebraic ones, such as Noether's Theorem, Hensel's Lemma, and the Weiestrass Preparation Theorem, as is brought out in detail in the book. Hensel's Lemma is useful in studying parameterizations since it characterizes when factorization in the polynomial ring can be "lifted" to a factorization in the formal power series ring. In this book Hensel's Lemma is used to study the irreducibility of what the authors define as `analytic curves' and `analytic cycles', which as the name implies, are algebraic curves defined in terms of the zero sets not of polynomials, but rather of formal power series. For readers who have had prior exposure to algebraic geometry, the way the authors use Hensel's Lemma also gives insight into what is called the `completion' of a ring using formal power series, and allows one to study the properties of an algebraic variety in neighborhoods that are smaller than Zariski open neighborhoods. The key idea here is the use of formal power series, in that they can give information about arbitrarily small neighborhoods of a node for example. In a Zariski neighborhood of a node for example, the curve may be irreducible, but in a smaller neighborhood, the use of formal power series reveals that the curve is reducible (due to the ability to factorize the relevant polynomial in the power series ring).
The concept of a `generic' point is discussed in the book no doubt as a prelude to the reader's possible future confrontation with the theory of schemes. The generic points should be thought of as not being singled out or special, and hence it is not surprising that an irreducible curve has infinitely many generic points (this is proved in the book). Generic points satisfy the polynomial in some proper extension of the field under consideration.
It is more difficult to grasp the relevance of a model of a function field, at least if one is not reminded of its role in the birational equivalence of algebraic curves. The authors don't motivate the concept too well, but eventually prove that any two models of a field are birationally equivalent. In the book, models and branch representations find utility in the concept of a `place' of a function field, which is an equivalence class of primitive place representations. The authors prove that the places and branches of any model of a function field are in a 1-1 correspondence.
Of particular importance is the authors' discussion of the notion of separability. In the theory of elliptic curves, one studies endomorphisms of elliptic curves, which are homomorphisms of elliptic curves to themselves that are given by rational functions. There is a standard form for the rational functions that describe an endomorphism of an elliptic curve y^2 = x^3 + Ax + B, namely as a pair (r(x), s(x)y) where r(x) and s(x) are rational functions. If the derivative of r(x) is not identically zero, then the endomorphism is said to be `separable.' As is shown in standard treatments of elliptic curves, the Frobenius endomorphism of an elliptic curve is not separable, and this turns out to be extremely useful property in the study of elliptic curves over finite fields. Separability becomes a sticking point when the authors discuss derivations and differentials, since an inseparable variable can be written as a power, and for fields of non-zero characteristic, the differential of such a variable will vanish.
Because of the drive towards generality in this book, the authors treat the notion of separable endomorphisms of algebraic curves in terms of derivatives, but first define separability in terms of field extensions. Their discussion is a straightforward summary of what is done in a graduate course in algebra (but the reader should be aware that the summary of separable field extensions is very cursory and most of it is delegated to an appendix). The important fact to remember here is that of a separable variable of a field of transcendence degree 1, which is one that cannot be written as the pth power of any element in this field. Using this notion of separability, one can generalize the Frobenius map of elliptic curves to that of a Frobenius rational transformation of the function field of an algebraic curve, which as expected takes two variables in this field and raises them to the qth power. Rational transformations are then defined as separable if the field extension of their images is separable; inseparable otherwise. The authors prove that every inseparable transformation is the product of a separable rational transformation and a Frobenius transformation. Characterizing separability of a rational transformation in terms of derivatives requires that differentiation is a meaningful operation on a field. This operation is called a `derivation', and the authors' define it in terms of its familiar properties, such as the product rule for differentiation etc. Things of course get tricky for higher derivatives in fields with non-zero characteristic, but this situation is dealt with by the use of `Hasse derivatives'.
The reader familiar with algebraic or differential topology will think of the genus as the number of "handles" on a surface or manifold, with an explicit formula usually given for its computation. This carries over to a large extent to algebraic geometry, but one has to worry about the singularities of the (irreducible) algebraic curve. The authors show how to do this in the context of places, and as expected, the Euler formula is generalized to a formula involving the order of a place, instead of the index of a critical point, as is usually done. The genus of an irreducible algebraic curve is then defined to be the genus of its function field. Defining the genus to this level of generality may seem like overkill to readers at this point, with the justification being fully explained later in the book in the context of the Riemann-Roch theorem.
Another very interesting discussion is the notions of the dual and bidual of an irreducible plane curve. As might be expected, these notions are similar to what is done in the theory of vector spaces. The dual curve of an algebraic curve is the irreducible curve consisting of points that form the equation of a tangent line to each non-singular point of the curve (singular points of course will not have unique tangent lines). The `rational Gauss map', defined early on in the section on dual curves, will when operating on the curve give the dual curve, as the authors show. The key point to be remembered here is that the Gauss map is a rational transformation, i.e. it may not be regular because of possible singularities in the curve. The authors give a coordinate expression for the Gauss map, defined in terms of the partial derivatives of the polynomial defining the curve. Readers familiar will elementary differential geometry will see sort of resemblance to the notion of the Gauss map in that field, if one remembers that the latter is defined in terms of normals to a curve, rather than tangents. The Gauss map also plays a major role in the theory of vector bundles, where it is used to assist in the classification of vector bundles. In all of these contexts, one will observe the key role played by projective space in defining the Gauss map. Taking the dual of the dual of a curve defines the `bidual' of a curve, which for fields of characteristic zero is the same as the curve itself. This fails in fields of positive characteristic, as the authors show in detail. Curves where the Gauss maps of the curve and its dual are separable are called `reflexive', and the authors prove a theorem that characterizes when a curve is bidual and when it is non-reflexive.
Going through the long series of definitions in the first section of Chapter 6 pays off in section 2 wherein the authors discuss how linear series arise from `linear systems of curves' and `intersection divisors', where the latter is defined in terms of two irreducible curves and the intersection multiplicity serves as the weight in the sum over all places. Again, a historical motivation would enable the reader a better appreciation of the discussion, such as for example why there is a difference in dimension between a linear system and a linear series. Some concrete examples of linear systems would also be helpful. Linear systems of curves go way back to the origins of algebraic geometry, particularly the work of the Italian geometers, and they have taken on particular importance in enumerative geometry due to the connections with physics via superstring theory.
The concept of a linear system of curves can be described intuitively by first choosing a collection of distinct points {p1,...,pn} in the plane, a list of nonnegative integers {m1,...mn}, and concentrating attention on the collection of plane curves of degree d that have a multiplicity of at least mi at pi. This collection is a linear subspace of the space of plane curves of degree and is called a `linear system' of plane curves. If F is an irreducible curve that is not contained as a component in this linear system, then the authors show that this linear system will "cut out" on F the divisors of a linear series. The authors' strategy for studying linear series is to choose a model so that F has only ordinary singularities, and then they define what is called the `adjoint' curve of F. They prove that an r-fold point of F is at least an (r-1)-fold point of the adjoint, and vice versa, and also prove that the adjoints of a given degree cut out a (complete) linear series on F. If restriction is placed on curves of degree m greater than 3, the adjoints of degree m - 3 are called the `canonical' adjoints of the curves. The authors prove that for a curve of degree greater than 3 and genus g, there are g canonical adjoints for the curve that are linearly independent. Readers can gain much further insight into linear systems and linear series by consulting some of the older books on algebraic curves, say around the beginning decades of the twentieth century. In some of those books (one of them is included in the collection of references), one will find that linear series are thought of as being those points of intersection of a collection of curves with a "base curve". The argumentation in these books is of course not as rigorous as encountered in this book, but it does serve to develop more of an appreciation behind the concept of a linear series and linear systems.
The authors prove the Riemann-Roch theorem in the context of divisors in this chapter, but readers who are familiar with the theory of algebraic curves over the complex numbers or meromorphic functions of a complex variable will see it reformulated using terminology that is closer to that of the "zeros and poles of functions". Such a background will also assist in the understanding of the Weierstrass gap theorem, which is also proven in this chapter, since in the context of meromorphic functions one is interested in finding poles of a particular order at a given point. Finding a pole at this order is dependent on the genus. In this book the authors prove that there are exactly g "gap numbers", i.e. an integer where the function does not have a pole of the order of this integer. If the genus is zero for example then the function can have a pole of any order at any point. If the genus is greater than or equal to one then a pole at a point will always have an order greater than the genus except at the `Weierstrass points.' For a given point, the g gap values are situated between 1 and 2*genus and have a strict ordering. Weierstrass gaps have recently become of interest in coding theory (some of which is discussed in the last chapter of the book). The authors also show how to put an irreducible curve of genus greater than or equal to 1 in `Weierstrass normal form', a strategy that should be very familiar to those readers with a background in elliptic curves.
For now, this review is based on a reading of Chapters 1-6 of the book.
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