Algebraic Geometry And Arithmetic Curves

  algebraic geometry and arithmetic curves: Algebraic Geometry and Arithmetic Curves Qing Liu, Reinie Erne, 2006-06-29 This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.
  algebraic geometry and arithmetic curves: Algebraic Geometry and Arithmetic Curves 刘擎, 2002 Based on the author's course for first-year students this well-written text explains how the tools of algebraic geometry and of number theory can be applied to a study of curves. The book starts by introducing the essential background material and includes 600 exercises.
  algebraic geometry and arithmetic curves: Algebraic Geometry and Arithmetic Curves Qing Liu, 2006-06-29 This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. This book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are few, and including many examples and approximately 600 exercises, the book is ideal for graduate students.
  algebraic geometry and arithmetic curves: Arithmetic Algebraic Geometry Brian David Conrad, The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.
  algebraic geometry and arithmetic curves: Arithmetic of Algebraic Curves Serguei A. Stepanov, 1994-12-31 Author S.A. Stepanov thoroughly investigates the current state of the theory of Diophantine equations and its related methods. Discussions focus on arithmetic, algebraic-geometric, and logical aspects of the problem. Designed for students as well as researchers, the book includes over 250 excercises accompanied by hints, instructions, and references. Written in a clear manner, this text does not require readers to have special knowledge of modern methods of algebraic geometry.
  algebraic geometry and arithmetic curves: Algebraic Geometry Robin Hartshorne, 2013-06-29 An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of Residues and Duality, Foundations of Projective Geometry, Ample Subvarieties of Algebraic Varieties, and numerous research titles.
  algebraic geometry and arithmetic curves: An Invitation to Arithmetic Geometry Dino Lorenzini, 2021-12-23 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. … 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 MATH The interaction between number theory and algebraic geometry has been especially fruitful. 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.
  algebraic geometry and arithmetic curves: Algebraic Geometry and Commutative Algebra Siegfried Bosch, 2022-04-22 Algebraic Geometry is a fascinating branch of Mathematics that combines methods from both Algebra and Geometry. It transcends the limited scope of pure Algebra by means of geometric construction principles. Putting forward this idea, Grothendieck revolutionized Algebraic Geometry in the late 1950s by inventing schemes. Schemes now also play an important role in Algebraic Number Theory, a field that used to be far away from Geometry. The new point of view paved the way for spectacular progress, such as the proof of Fermat's Last Theorem by Wiles and Taylor. This book explains the scheme-theoretic approach to Algebraic Geometry for non-experts, while more advanced readers can use it to broaden their view on the subject. A separate part presents the necessary prerequisites from Commutative Algebra, thereby providing an accessible and self-contained introduction to advanced Algebraic Geometry. Every chapter of the book is preceded by a motivating introduction with an informal discussion of its contents and background. Typical examples, and an abundance of exercises illustrate each section. Therefore the book is an excellent companion for self-studying or for complementing skills that have already been acquired. It can just as well serve as a convenient source for (reading) course material and, in any case, as supplementary literature. The present edition is a critical revision of the earlier text.
  algebraic geometry and arithmetic curves: Algebraic Geometry Solomon Lefschetz, 2012-09-05 An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
  algebraic geometry and arithmetic curves: Algebraic Curves and Riemann Surfaces Rick Miranda, 1995 In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.
  algebraic geometry and arithmetic curves: The Arithmetic of Elliptic Curves Joseph H. Silverman, 2013-03-09 The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.
  algebraic geometry and arithmetic curves: The Geometry of Schemes David Eisenbud, Joe Harris, 2006-04-06 Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.
  algebraic geometry and arithmetic curves: Arithmetic of Higher-Dimensional Algebraic Varieties Bjorn Poonen, Yuri Tschinkel, 2012-12-06 This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.
  algebraic geometry and arithmetic curves: Algebraic Geometry Ulrich Görtz, Torsten Wedhorn, 2010-08-06 This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.
  algebraic geometry and arithmetic curves: Algebraic Curves and Their Applications Lubjana Beshaj, Tony Shaska, 2019-02-26 This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide many applications in number theory, computer security and cryptography, coding theory, differential equations, and more. Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, hyperelliptic curves, dessins d'enfants, applications to Painlevé equations, descent on real algebraic varieties, quadratic residue codes based on hyperelliptic curves, and Abelian varieties and cryptography. This book will be a valuable resource for people interested in algebraic curves and their connections to other branches of mathematics.
  algebraic geometry and arithmetic curves: Rational Points on Elliptic Curves Joseph H. Silverman, John Tate, 2013-04-17 The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
  algebraic geometry and arithmetic curves: LMSST: 24 Lectures on Elliptic Curves John William Scott Cassels, 1991-11-21 A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.
  algebraic geometry and arithmetic curves: Vertex Algebras and Algebraic Curves Edward Frenkel, David Ben-Zvi, 2004-08-25 Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book contains many original results, introduces important new concepts, and brings new insights into the theory of vertex algebras. The authors have made a great effort to make the book self-contained and accessible to readers of all backgrounds. Reviewers of the first edition anticipated that it would have a long-lasting influence on this exciting field of mathematics and would be very useful for graduate students and researchers interested in the subject. This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.
  algebraic geometry and arithmetic curves: Complex Algebraic Curves Frances Clare Kirwan, 1992-02-20 This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.
  algebraic geometry and arithmetic curves: Commutative Algebra David Eisenbud, 2013-12-01 This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included.
  algebraic geometry and arithmetic curves: Rigid Geometry of Curves and Their Jacobians Werner Lütkebohmert, 2016-01-26 This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.
  algebraic geometry and arithmetic curves: Algebraic Curves William Fulton, 2008 The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites. We have assumed that the reader is familiar with some basic properties of rings, ideals and polynomials, such as is often covered in a one-semester course in modern algebra; additional commutative algebra is developed in later sections.
  algebraic geometry and arithmetic curves: Modular Forms and Fermat’s Last Theorem Gary Cornell, Joseph H. Silverman, Glenn Stevens, 2013-12-01 This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
  algebraic geometry and arithmetic curves: Elliptic Curves Henry McKean, Victor Moll, 1999-08-13 An introductory 1997 account in the style of the original discoverers, treating the fundamental themes even-handedly.
  algebraic geometry and arithmetic curves: Number Theory and Geometry: An Introduction to Arithmetic Geometry Álvaro Lozano-Robledo, 2019-03-21 Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. Similarly, Gauss's law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively. This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.
  algebraic geometry and arithmetic curves: Birational Geometry, Rational Curves, and Arithmetic Fedor Bogomolov, Brendan Hassett, Yuri Tschinkel, 2013-05-17 ​​​​This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.
  algebraic geometry and arithmetic curves: Geometry and Arithmetic Carel Faber, Gavril Farkas, Robin de Jong, 2012 This volume contains 21 articles written by leading experts in the fields of algebraic and arithmetic geometry. The treated topics range over a variety of themes, including moduli spaces of curves and abelian varieties, algebraic cycles, vector bundles and coherent sheaves, curves over finite fields, and algebraic surfaces, among others. The volume originates from the conference Geometry and Arithmetic, which was held on the island of Schiermonnikoog in The Netherlands in September 2010.
  algebraic geometry and arithmetic curves: Advances on Superelliptic Curves and Their Applications L. Beshaj, T. Shaska, E. Zhupa, 2015-07-16 This book had its origins in the NATO Advanced Study Institute (ASI) held in Ohrid, Macedonia, in 2014. The focus of this ASI was the arithmetic of superelliptic curves and their application in different scientific areas, including whether all the applications of hyperelliptic curves, such as cryptography, mathematical physics, quantum computation and diophantine geometry, can be carried over to the superelliptic curves. Additional papers have been added which provide some background for readers who were not at the conference, with the intention of making the book logically more complete and easier to read, but familiarity with the basic facts of algebraic geometry, commutative algebra and number theory are assumed. The book is divided into three sections. The first part deals with superelliptic curves with regard to complex numbers, the automorphisms group and the corresponding Hurwitz loci. The second part of the book focuses on the arithmetic of the subject, while the third addresses some of the applications of superelliptic curves.
  algebraic geometry and arithmetic curves: Algebraic Geometry in Coding Theory and Cryptography Harald Niederreiter, Chaoping Xing, 2009-09-21 This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books
  algebraic geometry and arithmetic curves: Galois Representations in Arithmetic Algebraic Geometry A. J. Scholl, Richard Lawrence Taylor, 1998-11-26 Conference proceedings based on the 1996 LMS Durham Symposium 'Galois representations in arithmetic algebraic geometry'.
  algebraic geometry and arithmetic curves: Algebra, Arithmetic, and Geometry Yuri Tschinkel, Yuri Zarhin, 2010-04-11 EMAlgebra, Arithmetic, and Geometry: In Honor of Yu. I. ManinEM consists of invited expository and research articles on new developments arising from Manin’s outstanding contributions to mathematics.
  algebraic geometry and arithmetic curves: Arithmetic Geometry G. Cornell, J. H. Silverman, 2012-12-06 This volume is the result of a (mainly) instructional conference on arithmetic geometry, held from July 30 through August 10, 1984 at the University of Connecticut in Storrs. This volume contains expanded versions of almost all the instructional lectures given during the conference. In addition to these expository lectures, this volume contains a translation into English of Falt ings' seminal paper which provided the inspiration for the conference. We thank Professor Faltings for his permission to publish the translation and Edward Shipz who did the translation. We thank all the people who spoke at the Storrs conference, both for helping to make it a successful meeting and enabling us to publish this volume. We would especially like to thank David Rohrlich, who delivered the lectures on height functions (Chapter VI) when the second editor was unavoidably detained. In addition to the editors, Michael Artin and John Tate served on the organizing committee for the conference and much of the success of the conference was due to them-our thanks go to them for their assistance. Finally, the conference was only made possible through generous grants from the Vaughn Foundation and the National Science Foundation.
  algebraic geometry and arithmetic curves: Lectures on Logarithmic Algebraic Geometry Arthur Ogus, 2018-11-08 A self-contained introduction to logarithmic geometry, a key tool for analyzing compactification and degeneration in algebraic geometry.
  algebraic geometry and arithmetic curves: Basic Algebraic Geometry 2 Igor Rostislavovich Shafarevich, 1994 The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with Volume 1 the author has revised the text and added new material, e.g. a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as in theoretical physics.
  algebraic geometry and arithmetic curves: Undergraduate Algebraic Geometry Miles Reid, Miles A. Reid, 1988-12-15 Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.
  algebraic geometry and arithmetic curves: Linear Algebra and Geometry Igor R. Shafarevich, Alexey O. Remizov, 2012-08-23 This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.
  algebraic geometry and arithmetic curves: Algebraic Geometry I: Schemes Ulrich Görtz, Torsten Wedhorn, 2020-07-27 This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.
  algebraic geometry and arithmetic curves: An Introduction to Invariants and Moduli Shigeru Mukai, 2003-09-08 Sample Text
  algebraic geometry and arithmetic curves: Algebraic Curves and Cryptography V. Kumar Murty,
  algebraic geometry and arithmetic curves: Algebraic Curves in Cryptography San Ling, Huaxiong Wang, Chaoping Xing, 2013-06-13 The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh
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(4) We will discuss various aspects of the geometry and arithmetic of surfaces as they arise, and some singularity theory and other topics according to interest. 2.1. References. • Beauville, …

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Algebraic geometry played a central role in 19th century math. The deepest results of Abel, Riemann, Weierstrass, and many of the most important works of Klein and Poincaré were part …

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Grothendieck referred to as the “Fundamental Conjecture of Anabelian Algebraic Geometry.” (GC1) “Fundamental Conjecture.” An anabelian algebraic variety X over a field K which is …

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Algebraic Geometry And Arithmetic Curves(1): Algebraic Geometry and Arithmetic Curves Qing Liu,Reinie Erne,2006-06-29 This book is a general introduction to the theory of schemes …

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Effective geometry and arithmetic of curves: an introduction Online CIMPA course ... II Arithmetic of curves and its Jacobian over finite fields ... 7.1 Abelian varieties: algebraic and complex …

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NOTES FOR A COURSE IN ALGEBRAIC GEOMETRY
Chapter 1: PLANE CURVES 1.1 The Affine Plane 1.2 The Projective Plane 1.3 Plane Projective Curves 1.4 Tangent Lines 1.5 digression: Transcendence Degree 1.6 The Dual Curve 1.7 …

PROCEEDING S OF SYMPOSIA - American Mathematical …
Summer Research Institute on Algebraic Geometry (1985: Bowdoin College) Algebraic geometry: Bowdoin 1985/Spencer J. Bloch, editor; with the collaboration of H. Clemens. . . [et al.]. …

The arithmetic and geometry of genus four curves - The …
The arithmetic and geometry of genus four curves Hang Xue ... whether they are torsion in the corresponding Chow group is an important problem in algebraic and arithmetic geometry. An …

Algebraic Geometry (Math 631) - Institute for Advanced Study
Algebraic geometry (AG) is the study of algebraic varieties, i.e., simultaneous zero loci of polynomial equations in multiple variables. It is one of the oldest subjects in mathematics, with …

Arithmetic Algebraic Geometry - GBV
Arithmetic Algebraic Geometry Brian Conrad Karl Rubin Editors (3) American Mathematical Society Institute for Advanced Study. Contents Preface dd ... An Introduction to the p-adic …

Algebraic Curves and Their Applications - American …
Local and Global Methods in Algebraic Geometry, 2018 711 Thomas Creutzig and Andrew R. Linshaw, Editors, Vertex Algebras and ... Papers cover topics such as the rational torsion …

Math 248B. Modular curves Contents M - Stanford University
18. Generalized elliptic curves 42 19. Tate curves 43 20. Moduli of generalized elliptic curves 45 1. Introduction We seek an \arithmetic" construction of the theory of modular curves and modular …

Algebraic Geometry Iii Complex Algebraic Varieties Algebraic …
Algebraic Geometry III A.N. Parshin,I.R. Shafarevich,2013-04-17 This two part EMS volume provides a succinct summary of complex algebraic geometry coupled with a lucid introduction …

ALGEBRAIC CURVES Contents - Columbia University
ALGEBRAIC CURVES 0BRV Contents 1. Introduction 1 2. Curvesandfunctionfields 2 3. Linearseries 5 4. Duality 7 5. Riemann-Roch 10 6. Somevanishingresults 12 7. …

GEOMETRY AND ARITHMETIC OF LOW GENUS CURVES …
GEOMETRY AND ARITHMETIC OF LOW GENUS CURVES 3 (b)Describe an a ne open covering of E. Show that the function eld of Eis given by kpx;yq y2 p x e 1qpx e 2qpx e 3q …

Riemann-Hurwitz and Applications - Columbia University
While the de nition above may seem a little bit odd to one unfamiliar with algebraic geometry, it should be clear why the picard group is important after the following proposition. Proposition 8. …

Torsion Points on Curves
Galois Representations and Arithmetic Algebraic Geometry pp. 235-247 Torsion Points on Curves Robert F. Coleman § 1. Let C be a smooth complete curve defined over a field K. Let K denote …

Algebraic Geometry Iii Complex Algebraic Varieties Algebraic …
Algebraic Geometry III A.N. Parshin,I.R. Shafarevich,2013-04-17 This two part EMS volume provides a succinct summary of complex algebraic geometry coupled with a lucid introduction …

Algebraic Geometry And Arithmetic Curves By Qing Liu
Algebraic Geometry And Arithmetic Curves By Qing Liu Marc Hindry,Joseph H. Silverman Algebraic Geometry and Arithmetic Curves Qing Liu,Reinie Erne,2006-06-29 This book is a …

Topics in Number Theory: Elliptic Curves - Department of …
Arithmetic geometry is the study of rational and integer points on algebraic varieties. Elliptic curves are, perhaps, the best place to start: there’s a great deal we know, and a great deal we …

Lecture Notes of C2.6 Introduction to schemes. - University …
A first course in algebraic geometry is desirable but not technically necessary. The terminology of these notes will generally be in line with the terminology of Hartshorne’s book [Har77]. ...

Math 259x: Moduli Spaces in Algebraic Geometry - Harvard …
One of the characterizing features of algebraic geometry is that the set of all geometric objects of a fixed type (e.g. smooth projective curves, subspaces of a fixed vector space, or coherent …

Birational geometry of moduli spaces of log surfaces.
My work focuses on the geometry and arithmetic of moduli spaces, particularly on moduli spaces of surfaces. In contrast to curves, the geometry of moduli spaces of surfaces is much less …

MATH 137 NOTES: UNDERGRADUATE ALGEBRAIC …
Algebraic geometry begins here. Goal 3.3. The goal of algebraic geometry is to relate the algebra of f to the geometry of its zero locus. This was the goal until the second decade of the …

Algebraic Geometry II - TU Darmstadt
Algebraic Geometry II Prof. Dr. Torsten Wedhorn, Prof. Dr. Timo Richarz Contact M.Sc. Can Yaylali Schlossgartenstraße 7, S2|15-313 yaylali (ergänze @mathematik.tu-darmstadt.de) …

Arithmetic, Geometry, Cryptography and Coding Theory
geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] – Abelian varieties of di-mension> 1[Seealso14Kxx]. |Numbertheory–Arithmeticalgebraicgeometry(Diophantine …

Geometry of Curves over a Discrete Valuation Ring - ALGANT
Required Algebraic Geometry We begin with some basic definitions and facts 1 which are necessary for study the geometry of curves. 1.1 Varieties and Curves Definition 1.1.1. A ne n …

Abelian Varieties - James Milne
Some knowledge of schemes and algebraic number theory will also be helpful. References. In addition to the references listed at the end, I refer to the following of my course notes: GT …

Publications of Shou-Wu Zhang - Princeton University
37.(with Xinyi Yuan) The arithmetic Hodge index theorem for adelic line bundles II, Manuscript, 2021, 28 pages, 38. Height pairings for algebraic cycles on the product of a curve and a sur …

The Weil Conjectures for Curves - Columbia University
Caleb Ji The Weil Conjectures for Curves Summer 2021 Note that d dt logZ(X;t) = P 1 r=0 N r+1t r. Then with some elementary manipulation, we can connect this zeta function with a possibly …

Algebraic Geometry - James Milne
2 Algebraic Sets 30 Definition of an algebraic set. 30; The Hilbert basis theorem. 31; The Zariski topology. 32; The Hilbert Nullstellensatz. 33; The correspondence between algebraic sets and …

Introduction to Algebraic Geometry - American Mathematical …
Contents vii §10.6.Factorization of birational regular maps of nonsingular surfaces 168 §10.7.Projectiveembeddingofnonsingularvarieties 170 §10.8.Complexmanifolds 175

Riemann surfaces and algebraic curves - Department of …
22 2. RIEMANN SURFACES AND ALGEBRAIC CURVES Given a real affine algebraic curve of degree d C := f0 = f(x,y) = f d(x,y)+ f d 1(x,y)+ + f0gˆR2 with f d not identically zero, we would …