Computational Aspects Of Algebraic Curves

Author : Tanush Shaska
Publisher : World Scientific Publishing Company Incorporated
Page : 272 pages
File Size : 53,9 Mb
Release : 2005
Category : Mathematics
ISBN : 9812564594

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Computational Aspects of Algebraic Curves by Tanush Shaska Pdf

1. Preface -- 2. Foreword by the editor -- 3. A new proof for the non-degeneracy of the Frey-Rück pairing and a connection to isogenics over the base field / E. F. Schaefer -- 4. Elliptic curve torsion points and division polynomials / I. A. Burhanuddin and M. A. Huang -- 5. Detecting complex multiplication / J. D. Achter -- 6. Simple numerical uniformatization of elliptic curves / M. Seppälä -- 7. On the moduli space of Klein four covers of the projective line / D. Glass and R. Pries -- 8. Field of moduli and field of definition for curves of genus 2 / G. Cardona and J. Quer -- 9. Explicit computation of Hurwitz spectra / R. Vogeler -- 10. Non-normal Belyi p-gonal surfaces / A. Wootton -- 11. Hyperelliptic curves of genus 3 with prescribed automorphism group / J. Gutierrez, D. Sevilla, and T. Shaska -- 12. Curves over finite fields with many points : an introduction / J. Voight -- 13. Hyperelliptic curves of genus 3 and 4 in characteristic 2 / Y. Demirbas -- 14. Modular representations on some Riemann-Roch spaces of modular curves X(N) / D. Joyner and A. Ksir -- 15. Genus two curves covering elliptic curves : a computational approach / T. Shaska -- 16. A question about Pic(X) as a G-module / D. Goldstein, R. Guralnick, and D. Joyner -- 17. Galois groups of prime degree polynomials with nonreal roots / A. Bialostocki and T. Shaska -- 18. Counting generating systems of a finite group from given conjugacy classes / R. Staszewski, H. Völklein, and G. Wiesend -- 19. Group action on genus 3 curves and their Weierstrass points / H. Babu and P. Venkataraman

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 53,9 Mb
Release : 2024-07-01
Category : Electronic
ISBN : 9789814479578

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Computational Aspects of Algebraic Curves by Anonim Pdf

Author : Zhibin Liang,Chandrakant Aribam
Publisher : Springer
Page : 95 pages
File Size : 40,5 Mb
Release : 2019-05-22
Category : Mathematics
ISBN : 9789811366642

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The Computational and Theoretical Aspects of Elliptic Curves by Zhibin Liang,Chandrakant Aribam Pdf

This volume presents a collection of results related to the BSD conjecture, based on the first two India-China conferences on this topic. It provides an overview of the conjecture and a few special cases where the conjecture is proved. The broad theme of the two conferences was “Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture”. The first was held at Beijing International Centre for Mathematical Research (BICMR) in December 2014 and the second was held at the International Centre for Theoretical Sciences (ICTS), Bangalore, India in December 2016. Providing a broad overview of the subject, the book is a valuable resource for young researchers wishing to work in this area. The articles have an extensive list of references to enable diligent researchers to gain an idea of the current state of art on this conjecture.

Author : Ron Donagi,Tony Shaska
Publisher : Cambridge University Press
Page : 537 pages
File Size : 54,5 Mb
Release : 2020-03-02
Category : Mathematics
ISBN : 9781108715775

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Integrable Systems and Algebraic Geometry by Ron Donagi,Tony Shaska Pdf

A collection of articles discussing integrable systems and algebraic geometry from leading researchers in the field.

Author : Bas Edixhoven,Jean-Marc Couveignes
Publisher : Princeton University Press
Page : 438 pages
File Size : 49,7 Mb
Release : 2011-05-31
Category : Mathematics
ISBN : 9781400839001

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Computational Aspects of Modular Forms and Galois Representations by Bas Edixhoven,Jean-Marc Couveignes Pdf

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Author : Ron Donagi,Tony Shaska
Publisher : Cambridge University Press
Page : 421 pages
File Size : 42,9 Mb
Release : 2020-04-02
Category : Mathematics
ISBN : 9781108715744

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Integrable Systems and Algebraic Geometry by Ron Donagi,Tony Shaska Pdf

A collection of articles discussing integrable systems and algebraic geometry from leading researchers in the field.

Author : Susanne Schmitt,Horst G. Zimmer
Publisher : Walter de Gruyter
Page : 378 pages
File Size : 55,5 Mb
Release : 2008-08-22
Category : Mathematics
ISBN : 9783110198010

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Elliptic Curves by Susanne Schmitt,Horst G. Zimmer Pdf

The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.

Author : H. Kisilevsky
Publisher : American Mathematical Soc.
Page : 195 pages
File Size : 49,8 Mb
Release : 1994-01-01
Category : Mathematics
ISBN : 0821869949

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Elliptic Curves and Related Topics by H. Kisilevsky Pdf

This book represents the proceedings of a workshop on elliptic curves held in St. Adele, Quebec, in February 1992. Containing both expository and research articles on the theory of elliptic curves, this collection covers a range of topics, from Langlands's theory to the algebraic geometry of elliptic curves, from Iwasawa theory to computational aspects of elliptic curves. This book is especially significant in that it covers topics comprising the main ingredients in Andrew Wiles's recent result on Fermat's Last Theorem.

Author : Erich Kaltofen,Stephen M. Watt
Publisher : Springer Science & Business Media
Page : 336 pages
File Size : 47,6 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781461396475

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Computers and Mathematics by Erich Kaltofen,Stephen M. Watt Pdf

Advances in computer technology have had a tremendous impact on mathematics in the last two decades. In June of 1989, an international conference was held at MIT, bringing together mathematicians and computer scientists, to survey the work that has been done in computational mathematics, to report recent results in this field, and to discuss research directions as well as educational issues. This book presents a fascinating collection of contributions on topics ranging from computational algebra, and parallel computing, to mathematics education. Mathematicians interested in the computational aspects of their discipline as well as computer scientists interested in mathematical applications will enjoy the integrative view provided by this book.

Author : Ernst Kunz
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 48,7 Mb
Release : 2007-06-10
Category : Mathematics
ISBN : 9780817644437

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Introduction to Plane Algebraic Curves by Ernst Kunz Pdf

* Employs proven conception of teaching topics in commutative algebra through a focus on their applications to algebraic geometry, a significant departure from other works on plane algebraic curves in which the topological-analytic aspects are stressed *Requires only a basic knowledge of algebra, with all necessary algebraic facts collected into several appendices * Studies algebraic curves over an algebraically closed field K and those of prime characteristic, which can be applied to coding theory and cryptography * Covers filtered algebras, the associated graded rings and Rees rings to deduce basic facts about intersection theory of plane curves, applications of which are standard tools of computer algebra * Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook

Author : Ciro Ciliberto,Friedrich Hirzebruch,Rick Miranda,Mina Teicher
Publisher : Springer Science & Business Media
Page : 343 pages
File Size : 47,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401010115

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Applications of Algebraic Geometry to Coding Theory, Physics and Computation by Ciro Ciliberto,Friedrich Hirzebruch,Rick Miranda,Mina Teicher Pdf

An up-to-date report on the current status of important research topics in algebraic geometry and its applications, such as computational algebra and geometry, singularity theory algorithms, numerical solutions of polynomial systems, coding theory, communication networks, and computer vision. Contributions on more fundamental aspects of algebraic geometry include expositions related to counting points on varieties over finite fields, Mori theory, linear systems, Abelian varieties, vector bundles on singular curves, degenerations of surfaces, and mirror symmetry of Calabi-Yau manifolds.

Author : David A. Cox,John Little,Donal O'Shea
Publisher : Springer
Page : 664 pages
File Size : 43,5 Mb
Release : 2015-04-30
Category : Mathematics
ISBN : 9783319167213

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Ideals, Varieties, and Algorithms by David A. Cox,John Little,Donal O'Shea Pdf

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of MapleTM, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used. Readers who are teaching from Ideals, Varieties, and Algorithms, or are studying the book on their own, may obtain a copy of the solutions manual by sending an email to [email protected]. From the reviews of previous editions: “...The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. ...The book is well-written. ...The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.” —Peter Schenzel, zbMATH, 2007 “I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.” —The American Mathematical Monthly

Author : Su Bu-qing,Liu Ding-yuan
Publisher : Elsevier
Page : 306 pages
File Size : 54,6 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483272283

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Computational Geometry by Su Bu-qing,Liu Ding-yuan Pdf

Computational Geometry: Curve and Surface Modeling provides information pertinent to the fundamental aspects of computational geometry. This book discusses the geometric properties of parametric polynomial curves by using the theory of affine invariants for algebraic curves. Organized into eight chapters, this book begins with an overview of the objects studies in computational geometry, namely surfaces and curves. This text then explores the developments in the theory and application of spline functions, which began with cubic spline functions. Other chapters consider the mechanical background of the cubic spline functions, which is the wooden spline with small deflection. This book discusses as well that in mathematical lofting the information of a geometric shape is given by a set of data points, while in geometric design other ways of representations are available. The final chapter deals with the concepts in the theory of algebraic curves. This book is a valuable resource for mathematicians.

Author : Lubjana Beshaj,Tony Shaska
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 51,8 Mb
Release : 2019-02-26
Category : Curves, Algebraic
ISBN : 9781470442477

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Algebraic Curves and Their Applications by Lubjana Beshaj,Tony Shaska Pdf

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.

Author : H. Kisilevsky,Maruti Ram Murty
Publisher : American Mathematical Soc.
Page : 208 pages
File Size : 54,6 Mb
Release : 1994-01-01
Category : Mathematics
ISBN : 0821870351

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Elliptic Curves and Related Topics by H. Kisilevsky,Maruti Ram Murty Pdf

This book represents the proceedings of a workshop on elliptic curves held in St. Adele, Quebec, in February 1992. Containing both expository and research articles on the theory of elliptic curves, this collection covers a range of topics, from Langlands's theory to the algebraic geometry of elliptic curves, from Iwasawa theory to computational aspects of elliptic curves. This book is especially significant in that it covers topics comprising the main ingredients in Andrew Wiles's recent result on Fermat's Last Theorem.