An Invitation to Arithmetic Geometry (Graduate Studies in Mathematics) by Dino Lorenzini (PDF)

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In this volume the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the one-dimensional case are proved, including Bombieri’s proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes which will aid the reader who goes to the next level of this rich subject.

User’s Reviews

Editorial Reviews: Review “Extremely carefully written, masterfully thought out, and skillfully arranged introduction … to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. Detailed discussions, full proofs, much effort at thorough motivations, a wealth of illustrating examples, numerous related exercises and problems, hints for further reading, and a rich bibliography characterize this text as an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject … a highly welcome addition to the existing literature.” — Zentralblatt fr Mathematik”In order to come straight to the point: this book represents an excellent introduction to Algebraic Number Theory and to Algebraic Curves as well by viewing both theories as part of Commutative Algebra … all proof are given in full detail and its concept as well thought-out.” — Monatshefte fr Mathematik

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Arithmetic geometry, a subject of vast importance both in mathematics and in applications such as cryptography and telecommunications is defined in this book as essentially the study of the solutions of polynomial equations in n variables with coefficients drawn from a ring, where this ring is typically the integers, the rational numbers, or the integers modulo a prime. The author gives the reader a fine overview of this subject that he views as a kind of “unification” of number theory, commutative algebra, and algebraic geometry.Most importantly, the author emphasizes that this “unification” is based on analogies between these areas. That these are only analogies will become readily apparent as the reader is exposed to such concepts as the dimension of a ring and the tangent line of an algebraic curve. The dimension of a ring is not analogous to the notion of the cardinality of a basis of a vector space, as some readers might expect, but rather is analogous to a measure of the complexity of the ring. Fields are considered to be “simple (i.e. have dimension zero) while principal ideal domains are more complex in that they have dimension equal to one. On the other hand, the concept of a tangent line to an affine curve is very analogous to what is found in elementary calculus, involving taking the familiar partial derivatives.The author gives historical motivations throughout the book, especially for such core ideas as the integral closure and the ring of integers. Readers are expected to approach the book with a good background in modern algebra, with familiarity with such concepts as field extensions and the field of fractions. The field of fractions is exemplified in the book for example as the field of rational functions of the ring of polynomials in one variable.The integral closure of a ring is motivated as a ring that arithmetical properties that are “most like” the ring in question, with emphasis based on understanding the roots of monic polynomials. If the ring is a principal ideal domain (PID), then its integral closure is not necessarily a PID, but it does have the property of unique factorization of ideals. Also, ideals in the integral closure of a PID are finitely generated. This motivates the notion of a Noetherian ring, wherein every ideal is finitely generated. Thus Noetherian rings are analogous to vector spaces with a basis, and this fact makes them much easier to study.A “geometrical” analog of integral closure is obtained by studying nonsingular plane curves. In algebraic geometry a plane curve is defined by an irreducible polynomial in two variables, where the polynomial is monic in one of the variables. Taking the integral closure of the ring of polynomials in the remaining variable in the function field also gives this curve. The connection between the ring of functions and the concept of integral closure is illustrated very clearly in this book by first looking at injective maps from the field of fractions of the ring of polynomial functions on the plane curve. The author shows that this latter ring is integrally closed if and only if it is a Dedekind domain. From the standpoint of geometry, this ring is integrally closed if and only if the plane curve is nonsingular. The author gives some helpful examples illustrating this.In elementary calculus and in more advanced fields of mathematics such as differential topology, singular points are detected by the use of derivatives. In this book, and in algebraic geometry in general, singular points are detected by using the concept of localization. A point of the curve will be nonsingular if and only if the ring of rational functions that are defined at the point in question is a principal ideal domain.Localization is a straightforward concept to understand, and even more so due to the clarity of the author’s presentation. Motivating it by the example of the rational numbers as arising from the integers by adjoining the inverses of all non-zero integers, the real interest lies in local rings, i.e. those rings that have a unique maximal ideal. In particular, the localization of a ring of functions at a maximal ideal has a geometric interpretation, namely the maximal ideal corresponds to a point, and the ring of functions localized at this point is the ring of rational functions defined at this point. The point is a nonsingular point if and only if the maximal ideal of the ring of functions is generated by one element.Noted in the examples of integral closure and singular points in plane curves is that the integral closure can be viewed as a “removal” of the singular point. This is usually referred to as “resolution of singularities” in more advanced texts and the actual process of removing the singularity is called “blowing up”. The author discusses this process briefly in the book. Readers familiar with the theory of differential equations will recognize the process as being very familiar to the use of Puiseux series to solve these equations. Historically, Isaac Newton introduced the first algorithm for doing resolution of singularities in the context solving equations in two variables. His algorithm has been extensively generalized in the context of the theory of resolution of singularities in algebraic geometry.An entire chapter of the book is devoted to determining which domains have the property of unique factorization of ideals. He proves that such a domain must be Noetherian, Dedekind, and have dimension equal to one. The factorization into ideals is of course a generalization of the usual fact that any integer has a unique factorization into primes. Of great importance in this discussion is the notion of ramification, which will be familiar to those readers acquainted with the theory of Riemann surfaces. The author brings out nicely the fact that ramification in the context of factorization has a geometric interpretation that is very similar to the one in the theory of Riemann surfaces.Also discussed in some detail in the chapter on factorization, which as the author remarks is very important in a subject called class field theory, is how primes factorize in various ring or field extensions. For the case of a Galois extension for example, the ramification indices are equal, as are the residual degrees (the residual degree is the degree of the field extension of the integers modulo a prime p by the algebraic number field modulo the prime ideal “above” p). The Galois group will act transitively on prime ideals “above” p. This notion of being “above” p, is made clearer by his construction of a map from the spectrum of the integral closure B of a Dedekind domain to the spectrum of the Dedekind domain itself. If the latter is denoted by A, then if P is a prime ideal in Spec(A), then the inverse mal of P is a collection of maximal ideals which can be viewed as being “above” P.In the context of this discussion, and of great importance in areas such as the Langlands program, is the notion of the `decomposition group’ of a maximal ideal M, which is essentially all elements of the Galois group that have M as a fixed point. The author devotes a chapter to this subject to this in the context of discriminants. Also of interest is a normal subgroup of the decomposition group called the `inertia group.’ It is the kernel of a map from the decomposition group to the Galois group of B/M over A/P. The maximal ideal M is ramified over A if and only if the inertia group is not the identity. The author proves the fundamental result that a prime ideal is ramified in a number field if and only if this prime ideal divides the discriminant of the number field.Zeta functions, which not only arise in arithmetic geometry and number theory but also in areas of physics such as quantum field theory, statistical mechanics, and wireless communication systems, are studied in Chapter 8. Of particular interest is the Dedekind zeta function of a number field K, and the author shows how this function is constructed from local information at each prime ideal, how global information on the corresponding ring of integers K can be obtained from the Dedekind zeta function from its residue at a simple pole, and to what extent the Dedekind zeta function determines K (it does not uniquely, but it does allow the determination of the minimal Galois extension of the rational numbers Q that contains K). The author also shows how to relate the zeta function of a projective curve to its associated affine curve. The most valuable discussion is this chapter is how to express the Riemann hypothesis for curves over finite fields to a simple expression involving the zeroes of the zeta function of a nonsingular plane projective curve. This discussion sets the stage for the eventual proof of this hypothesis in Chapter 10.There are many topics of great importance in Chapter 9, which discusses the Riemann-Roch theorem, and the author gives the reader a brief sampling of sheaf theory, the latter of which is of great importance and power. The Riemann-Roch theorem is basically a result that relates the number of poles and the number of zeros of nonconstant (rational) functions on a nonsingular complete curve. The author’s treatment of this theorem is interesting in that he uses the language of valuations and the valuation ring to motivate it. It is quite amazing that one can reach the generality that the author does in this chapter without using elaborate mathematical machinery, even though he restricts himself to curves. The author remarks that the proofs could be considerably simplified if sheaf theory is used. It is the opinion of the reviewer that the reader’s intuition would probably be sacrificed if this were done at this level.Chapter 10, which discusses the famous Riemann hypothesis is very useful in that it gives a detailed discussion of the Bombieri proof of the Riemann hypothesis for nonsingular complete curves over a finite field (the case for the complex number field is still an outstanding problem). The inclusion of this proof is convenient in that it is understandable and concise, saving the reader considerable time which otherwise might be spent combing through the literature in order to distill the proof to manageable length. In discussing the existence of bounds for the number of rational points on curves over finite fields, the author points to the topic of desingularization of algebraic curves. This topic is of extreme importance to applications of algebraic and arithmetic geometry to such areas as error-correcting codes, bifurcation theory, and cryptography. The desingularization also allows a definition of the genus for singular curves, and the author devotes a few sentences alerting the reader to the fact that the Riemann-Roch theorem does hold for singular curves if the geometric genus is replaced by the arithmetic genus.Although short, the last chapter on “Further topics” gives and effective overview of some of the exciting developments in arithmetic geometry that were in play at the time of publication of this book (1996). The Langlands program are mentioned, and this area, which could be viewed as a grand attempt to follow the Weil program of unification of number fields and function fields has been subjected to considerable development since the time of publication. Readers interested in conducting research in this area will find this book ample preparation. In Arakelov geometry in particular, readers will put to good use the author’s introduction to valuation theory in this book. Arakelov theory allows one to do a kind of “unified arithmetic geometry” wherein both non-Archimedean and Archimedean valuations are treated on the same footing. A version of the Riemann-Roch theorem for arithmetic curves can be accomplished in the context of Arakelov theory, and the author spends a few pages motivating how this could be done in the context of number fields, and his summary is extremely helpful.

⭐Un libro que introduce al estudiante en la Teoría de números algebraicos y en las aplicaciones de la Geometría Algebraica en este campo, solo se requiere que el lector tenga los conocimientos de un curso de Algebra Abstracta.

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