Heegner Modules And Elliptic Curves
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Author | : Martin L. Brown |
Publisher | : Springer |
Total Pages | : 523 |
Release | : 2004-08-30 |
Genre | : Mathematics |
ISBN | : 3540444750 |
Heegner points on both modular curves and elliptic curves over global fields of any characteristic form the topic of this research monograph. The Heegner module of an elliptic curve is an original concept introduced in this text. The computation of the cohomology of the Heegner module is the main technical result and is applied to prove the Tate conjecture for a class of elliptic surfaces over finite fields, this conjecture is equivalent to the Birch and Swinnerton-Dyer conjecture for the corresponding elliptic curves over global fields.
Author | : |
Publisher | : Springer Science & Business Media |
Total Pages | : 532 |
Release | : 2004 |
Genre | : Algebraic fields |
ISBN | : 9783540222903 |
Author | : Henri Darmon |
Publisher | : Cambridge University Press |
Total Pages | : 386 |
Release | : 2004-06-21 |
Genre | : Mathematics |
ISBN | : 9780521836593 |
Thirteen articles by leading contributors on the history of the Gross-Zagier formula and its developments.
Author | : H. Kisilevsky |
Publisher | : American Mathematical Soc. |
Total Pages | : 208 |
Release | : 1994-01-01 |
Genre | : Mathematics |
ISBN | : 9780821870358 |
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 | : Henri Darmon |
Publisher | : American Mathematical Soc. |
Total Pages | : 148 |
Release | : |
Genre | : Mathematics |
ISBN | : 9780821889459 |
The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.
Author | : Neal I. Koblitz |
Publisher | : Springer Science & Business Media |
Total Pages | : 262 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461209099 |
The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.
Author | : Susanne Schmitt |
Publisher | : Walter de Gruyter |
Total Pages | : 378 |
Release | : 2008-08-22 |
Genre | : Mathematics |
ISBN | : 3110198010 |
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 | : J. E. Cremona |
Publisher | : CUP Archive |
Total Pages | : 388 |
Release | : 1997-05-15 |
Genre | : Mathematics |
ISBN | : 9780521598200 |
This book presents an extensive set of tables giving information about elliptic curves.
Author | : N. Koblitz |
Publisher | : Springer Science & Business Media |
Total Pages | : 258 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1468402552 |
This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses, thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work.
Author | : Joseph H. Silverman |
Publisher | : Springer Science & Business Media |
Total Pages | : 482 |
Release | : 2013-12-01 |
Genre | : Mathematics |
ISBN | : 1461208513 |
In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.