Iwasawa Theory Of Elliptic Curves With Complex Multiplication
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Arithmetic Theory of Elliptic Curves
| Author | : J. Coates |
| Publisher | : Springer |
| Total Pages | : 269 |
| Release | : 2006-11-14 |
| Genre | : Mathematics |
| ISBN | : 3540481605 |
This volume contains the expanded versions of the lectures given by the authors at the C.I.M.E. instructional conference held in Cetraro, Italy, from July 12 to 19, 1997. The papers collected here are broad surveys of the current research in the arithmetic of elliptic curves, and also contain several new results which cannot be found elsewhere in the literature. Owing to clarity and elegance of exposition, and to the background material explicitly included in the text or quoted in the references, the volume is well suited to research students as well as to senior mathematicians.
Arithmetic Theory of Elliptic Curves
| Author | : J. Coates |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 276 |
| Release | : 1999-10-19 |
| Genre | : Mathematics |
| ISBN | : 9783540665465 |
This volume contains the expanded versions of the lectures given by the authors at the C.I.M.E. instructional conference held in Cetraro, Italy, from July 12 to 19, 1997. The papers collected here are broad surveys of the current research in the arithmetic of elliptic curves, and also contain several new results which cannot be found elsewhere in the literature. Owing to clarity and elegance of exposition, and to the background material explicitly included in the text or quoted in the references, the volume is well suited to research students as well as to senior mathematicians.
Advanced Topics in the Arithmetic of Elliptic Curves
| 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.
Elliptic Curves, Modular Forms and Iwasawa Theory
| Author | : David Loeffler |
| Publisher | : Springer |
| Total Pages | : 494 |
| Release | : 2017-01-15 |
| Genre | : Mathematics |
| ISBN | : 3319450328 |
Celebrating one of the leading figures in contemporary number theory – John H. Coates – on the occasion of his 70th birthday, this collection of contributions covers a range of topics in number theory, concentrating on the arithmetic of elliptic curves, modular forms, and Galois representations. Several of the contributions in this volume were presented at the conference Elliptic Curves, Modular Forms and Iwasawa Theory, held in honour of the 70th birthday of John Coates in Cambridge, March 25-27, 2015. The main unifying theme is Iwasawa theory, a field that John Coates himself has done much to create. This collection is indispensable reading for researchers in Iwasawa theory, and is interesting and valuable for those in many related fields.
Iwasawa Theory and Its Perspective, Volume 2
| Author | : Tadashi Ochiai |
| Publisher | : American Mathematical Society |
| Total Pages | : 228 |
| Release | : 2024-04-25 |
| Genre | : Mathematics |
| ISBN | : 1470456737 |
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.
Elementary Modular Iwasawa Theory
| Author | : Haruzo Hida |
| Publisher | : World Scientific |
| Total Pages | : 446 |
| Release | : 2021-10-04 |
| Genre | : Mathematics |
| ISBN | : 9811241384 |
This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.
Rational Points on Modular Elliptic Curves
| Author | : Henri Darmon |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 146 |
| Release | : 2004 |
| Genre | : Mathematics |
| ISBN | : 0821828681 |
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.
Elliptic Curves and Big Galois Representations
| Author | : Daniel Delbourgo |
| Publisher | : Cambridge University Press |
| Total Pages | : 283 |
| Release | : 2008-07-31 |
| Genre | : Mathematics |
| ISBN | : 0521728665 |
Describes the arithmetic of modular forms and elliptic curves; self-contained and ideal for both graduate students and professional number theorists.
