Arithmetic Geometry

Author: Fabrizio Catanese

Publisher: Cambridge University Press

ISBN: 9780521591331

Category: Mathematics

Page: 312

View: 4308

Brought together in this book are papers from a conference on arithmetic geometry held in Cortona. The contributions are from many of the leading authorities in this field and together they cover a wide spectrum of topics that give an unsurpassed overview of research into number theory, geometry and their interactions. All whose research interests lie in these areas will find much of interest in this volume.
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Arithmetic Geometry

Author: G. Cornell,J. H. Silverman

Publisher: Springer Science & Business Media

ISBN: 1461386551

Category: Mathematics

Page: 353

View: 8613

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.
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Arithmetic Geometry

Conference on Arithmetic Geometry with an Emphasis on Iwasawa Theory, March 15-18, 1993, Arizona State University

Author: Nancy Childress,John W. Jones

Publisher: American Mathematical Soc.

ISBN: 0821851748

Category: Mathematics

Page: 220

View: 8490

This book resulted from a research conference in arithmetic geometry held at Arizona State University in March 1993. The papers describe important recent advances in arithmetic geometry. Several articles deal with $p$-adic modular forms of half-integral weight and their roles in arithmetic geometry. The volume also contains material on the Iwasawa theory of cyclotomic fields, elliptic curves, and function fields, including $p$-adic $L$-functions and $p$-adic height pairings. Other articles focus on the inverse Galois problem, fields of definition of abelian varieties with real multiplication, and computation of torsion groups of elliptic curves. The volume also contains a previously unpublished letter of John Tate, written to J.-P. Serre in 1973, concerning Serre's conjecture on Galois representations. With contributions by some of the leading experts in the field, this book provides a look at the state of the art in arithmetic geometry.
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Arithmetic Geometry

Clay Mathematics Institute Summer School, Arithmetic Geometry, July 17-August 11, 2006, Mathematisches Institut, Georg-August-Universität, Göttingen, Germany

Author: Clay Mathematics Institute. Summer School

Publisher: American Mathematical Soc.

ISBN: 0821844768

Category: Mathematics

Page: 562

View: 7110

This book is based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen. Intended for graduate students and recent Ph.D.'s, this volume will introduce readers to modern techniques and outstanding conjectures at the interface of number theory and algebraic geometry. The main focus is rational points on algebraic varieties over non-algebraically closed fields. Do they exist? If not, can this be proven efficiently and algorithmically? When rational points do exist, are they finite in number and can they be found effectively? When there are infinitely many rational points, how are they distributed? For curves, a cohesive theory addressing these questions has emerged in the last few decades. Highlights include Faltings' finiteness theorem and Wiles's proof of Fermat's Last Theorem. Key techniques are drawn from the theory of elliptic curves, including modular curves and parametrizations, Heegner points, and heights. The arithmetic of higher-dimensional varieties is equally rich, offering a complex interplay of techniques including Shimura varieties, the minimal model program, moduli spaces of curves and maps, deformation theory, Galois cohomology, harmonic analysis, and automorphic functions. However, many foundational questions about the structure of rational points remain open, and research tends to focus on properties of specific classes of varieties.
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Arithmetic Geometry

Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007

Author: Jean-Louis Colliot-Thélène,Peter Swinnerton-Dyer,Paul Vojta

Publisher: Springer

ISBN: 3642159451

Category: Mathematics

Page: 232

View: 6360

Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties through arbitrary rings, in particular through non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene, Peter Swinnerton Dyer and Paul Vojta.
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Arithmetic, Geometry, and Coding Theory

Proceedings of the International Conference held at Centre International de Rencontres de Mathématiques (CIRM), Luminy, France, June 28 - July 2, 1993

Author: R. Pellikaan,M. Perret,S.G. Vladut

Publisher: Walter de Gruyter

ISBN: 3110811057

Category: Mathematics

Page: 300

View: 8483

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
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An Invitation to Arithmetic Geometry

Author: Dino Lorenzini

Publisher: American Mathematical Soc.

ISBN: 0821802674

Category: Arithmetical algebraic geometry

Page: 397

View: 3235

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.
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Arithmetic, Geometry, Cryptography and Coding Theory

13th Conference [on] Arithmetic, Geometry, Cryptography and Coding Theory, CIRM, Marseille, France, March 14-18, 2011 : Geocrypt 2011, Bastia, France, June 19-24, 2011

Author: Yves Aubry,Christophe Ritzenthaler,Alexey Zykin

Publisher: American Mathematical Soc.

ISBN: 0821875728

Category: Mathematics

Page: 183

View: 5641

This volume contains the proceedings of the 13th $\mathrm{AGC^2T}$ conference, held March 14-18, 2011, in Marseille, France, together with the proceedings of the 2011 Geocrypt conference, held June 19-24, 2011, in Bastia, France. The original research articles contained in this volume cover various topics ranging from algebraic number theory to Diophantine geometry, curves and abelian varieties over finite fields and applications to codes, boolean functions or cryptography. The international conference $\mathrm{AGC^2T}$, which is held every two years in Marseille, France, has been a major event in the area of applied arithmetic geometry for more than 25 years.
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Arithmetic, Geometry, Cryptography, and Coding Theory 2009

12th Conference on Arithmetic, Geometry, Cryptography, and Coding Theory, March 30-April 3, 2009, Marseille, France : Geocrypt Conference, April 27-May 1, 2009, Pointe-à-Pitre, Guadeloupe, France : European Science Foundation Exploratory Workshop [on] Curves, Coding Theory, and Cryptography, March 25-29, 2009, Marseille, France

Author: David R. Kohel,Robert Rolland

Publisher: American Mathematical Soc.

ISBN: 0821849557

Category: Mathematics

Page: 166

View: 6235

This volume contains the proceedings of the 12th conference on Arithmetic, Geometry, cryptography and coding Theory, held in Marseille, France from March 30 to April 3, 2009, as well as the first Geocrypt conference, held in pointe-a-pitre, guadeloupe, from April 27 to may 1, 2009, and the European science Foundation exploratory workshop on curves, coding Theory, and Cryptography, held in Marseille, France from March 25 to 29, 2009. The articles Contained in this volume come from three related symposia organized by the group Arithmetique et Theorie de I' Information in Marseille. The topics cover arithmetic properties of curves and higher dimensional varieties with applications to codes and cryptography.
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Arithmetic Geometry over Global Function Fields

Author: Gebhard Böckle,David Burns,David Goss,Dinesh Thakur,Fabien Trihan,Douglas Ulmer

Publisher: Springer

ISBN: 3034808534

Category: Mathematics

Page: 337

View: 3629

This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.
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Arithmetic, Geometry, Cryptography, and Coding Theory

International Conference, November 5-9, 2007, CIRM, Marseilles, France

Author: Gilles Lachaud,Christophe Ritzenthaler,Michael A. Tsfasman

Publisher: American Mathematical Soc.

ISBN: 0821847163

Category: Mathematics

Page: 206

View: 484

This volume contains the proceedings of the 11th conference on $\mathrm{AGC^{2}T}$, held in Marseille, France in November 2007. There are 12 original research articles covering asymptotic properties of global fields, arithmetic properties of curves and higher dimensional varieties, and applications to codes and cryptography. This volume also contains a survey article on applications of finite fields by J.-P. Serre. $\mathrm{AGC^{2}T}$ conferences take place in Marseille, France every 2 years. These international conferences have been a major event in the area of applied arithmetic geometry for more than 20 years.
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Computational Arithmetic Geometry

AMS Special Session on Computational Arithmetic Geometry, April 29-30, 2006, San Francisco State University, San Francisco, CA

Author: Kristin Estella Lauter,Kenneth Ribet

Publisher: American Mathematical Soc.

ISBN: 0821843206

Category: Mathematics

Page: 129

View: 2541

With the recent increase in available computing power, new computations are possible in many areas of arithmetic geometry. To name just a few examples, Cremona's tables of elliptic curves now go up to conductor 120,000 instead of just conductor 1,000, tables of Hilbert class fields are known for discriminant up to at least 5,000, and special values of Hilbert and Siegel modular forms can be calculated to extremely high precision. In many cases, these experimental capabilities have led to new observations and ideas for progress in the field. They have also led to natural algorithmic questions on the feasibility and efficiency of many computations, especially for the purpose of applications in cryptography. The AMS Special Session on Computational Arithmetic Geometry, held on April 29-30, 2006, in San Francisco, CA, gathered together many of the people currently working on the computational and algorithmic aspects of arithmetic geometry. This volume contains research articles related to talks given at the session. The majority of articles are devoted to various aspects of arithmetic geometry, mainly with a computational approach.
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The Large Sieve and its Applications

Arithmetic Geometry, Random Walks and Discrete Groups

Author: E. Kowalski

Publisher: Cambridge University Press

ISBN: 1139472976

Category: Mathematics

Page: N.A

View: 6751

Among the modern methods used to study prime numbers, the 'sieve' has been one of the most efficient. Originally conceived by Linnik in 1941, the 'large sieve' has developed extensively since the 1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.
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Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

Author: Andreas Malmendier,Tony Shaska

Publisher: American Mathematical Soc.

ISBN: 1470428563

Category: Arithmetical algebraic geometry

Page: 222

View: 2576

This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, Washington. Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics. The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic 3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
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Noncommutative Geometry, Arithmetic, and Related Topics

Proceedings of the Twenty-First Meeting of the Japan-U.S. Mathematics Institute

Author: Caterina Consani,Alain Connes

Publisher: JHU Press

ISBN: 1421403528

Category: Mathematics

Page: 311

View: 365

This valuable collection of essays by some of the world’s leading scholars in mathematics presents innovative and field-defining work at the intersection of noncommutative geometry and number theory. The interplay between these two fields centers on the study of the rich structure of the adele class space in noncommutative geometry, an important geometric space known to support and provide a geometric interpretation of the Riemann Weil explicit formulas in number theory. This space and the corresponding quantum statistical dynamical system are fundamental structures in the field of noncommutative geometry. Several papers in this volume focus on the "field with one element" subject, a new topic in arithmetic geometry; others highlight recent developments in noncommutative geometry, illustrating unexpected connections with tropical geometry, idempotent analysis, and the theory of hyper-structures in algebra. Originally presented at the Twenty-First Meeting of the Japan-U.S. Mathematics Institute, these essays collectively provide mathematicians and physicists with a comprehensive resource on the topic.
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Arithmetic and Geometry

Author: Luis Dieulefait,Gerd Faltings,D. R. Heath-Brown,Yuri I. Manin,B. Z. Moroz,Yu. V. Manin,Jean-Pierre Wintenberger

Publisher: Cambridge University Press

ISBN: 1107462541

Category: Mathematics

Page: 550

View: 7150

The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.
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Arithmetic, Geometry, Cryptography and Coding Theory

Author: Alp Bassa,Alain Couvreur,David Kohel

Publisher: American Mathematical Soc.

ISBN: 1470428105

Category: Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Finite ground fields

Page: 199

View: 5158

This volume contains the proceedings of the 15th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held at the Centre International de Rencontres Mathématiques in Marseille, France, from May 18–22, 2015. Since the first meeting almost 30 years ago, the biennial AGCT meetings have been one of the main events bringing together researchers interested in explicit aspects of arithmetic geometry and applications to coding theory and cryptography. This volume contains original research articles reflecting recent developments in the field.
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Arithmetic Geometry

Conference on Arithmetic Geometry with an Emphasis on Iwasawa Theory, March 15-18, 1993, Arizona State University

Author: Nancy Childress,John W. Jones

Publisher: American Mathematical Soc.

ISBN: 0821851748

Category: Mathematics

Page: 220

View: 3326

This book resulted from a research conference in arithmetic geometry held at Arizona State University in March 1993. The papers describe important recent advances in arithmetic geometry. Several articles deal with $p$-adic modular forms of half-integral weight and their roles in arithmetic geometry. The volume also contains material on the Iwasawa theory of cyclotomic fields, elliptic curves, and function fields, including $p$-adic $L$-functions and $p$-adic height pairings. Other articles focus on the inverse Galois problem, fields of definition of abelian varieties with real multiplication, and computation of torsion groups of elliptic curves. The volume also contains a previously unpublished letter of John Tate, written to J.-P. Serre in 1973, concerning Serre's conjecture on Galois representations. With contributions by some of the leading experts in the field, this book provides a look at the state of the art in arithmetic geometry.
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