Rational Points on Modular Elliptic Curves - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Rational Points on Modular Elliptic Curves write by Henri Darmon. This book was released on 2004. Rational Points on Modular Elliptic Curves available in PDF, EPUB and Kindle. 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.
Rational Points on Elliptic Curves
Rational Points on Elliptic Curves - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Rational Points on Elliptic Curves write by Joseph H. Silverman. This book was released on 2013-04-17. Rational Points on Elliptic Curves available in PDF, EPUB and Kindle. The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
Elliptic Curves, Modular Forms, and Their L-functions
Elliptic Curves, Modular Forms, and Their L-functions - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Elliptic Curves, Modular Forms, and Their L-functions write by Álvaro Lozano-Robledo. This book was released on 2011. Elliptic Curves, Modular Forms, and Their L-functions available in PDF, EPUB and Kindle. Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.
Algorithms for Modular Elliptic Curves Full Canadian Binding
Algorithms for Modular Elliptic Curves Full Canadian Binding - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Algorithms for Modular Elliptic Curves Full Canadian Binding write by J. E. Cremona. This book was released on 1997-05-15. Algorithms for Modular Elliptic Curves Full Canadian Binding available in PDF, EPUB and Kindle. This book presents an extensive set of tables giving information about elliptic curves.
Elliptic Curves (Second Edition)
Elliptic Curves (Second Edition) - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Elliptic Curves (Second Edition) write by James S Milne. This book was released on 2020-08-20. Elliptic Curves (Second Edition) available in PDF, EPUB and Kindle. This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses.An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer.Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work.The first three chapters develop the basic theory of elliptic curves.For this edition, the text has been completely revised and updated.
