Cm Points On Products Of Drinfeld Modular Curves
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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.
Algorithmic Number Theory
Author | : Joe P. Buhler |
Publisher | : Springer Science & Business Media |
Total Pages | : 660 |
Release | : 1998-06-05 |
Genre | : Computers |
ISBN | : 9783540646570 |
The field of diagnostic nuclear medicine has changed significantly during the past decade. This volume is designed to present the student and the professional with a comprehensive update of recent developments not found in other textbooks on the subject. The various clinical applications of nuclear medicine techniques are extensively considered, and due attention is given also to radiopharmaceuticals, equipment and instrumentation, reconstruction techniques and the principles of gene imaging.
Some Problems of Unlikely Intersections in Arithmetic and Geometry
Author | : Umberto Zannier |
Publisher | : Princeton University Press |
Total Pages | : 176 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 069115371X |
This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the Andr -Oort conjecture (outlining work by Pila).
Algebraic Cycles and Motives: Volume 1
Author | : Jan Nagel |
Publisher | : Cambridge University Press |
Total Pages | : 293 |
Release | : 2007-05-03 |
Genre | : Mathematics |
ISBN | : 0521701740 |
This 2007 book is a self-contained account of the subject of algebraic cycles and motives.
Arithmetic Geometry
Author | : Clay Mathematics Institute. Summer School |
Publisher | : American Mathematical Soc. |
Total Pages | : 570 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 0821844768 |
Based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen, this tile is intended for graduate students and recent PhD's. It introduces readers to modern techniques and conjectures at the interface of number theory and algebraic geometry.
Modular Forms, a Computational Approach
Author | : William A. Stein |
Publisher | : American Mathematical Soc. |
Total Pages | : 290 |
Release | : 2007-02-13 |
Genre | : Mathematics |
ISBN | : 0821839608 |
This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory. --John E. Cremona, University of Nottingham William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics.
Abstracts of Papers Presented to the American Mathematical Society
Author | : American Mathematical Society |
Publisher | : |
Total Pages | : 666 |
Release | : 2002 |
Genre | : Mathematics |
ISBN | : |