LMSST: 24 Lectures on Elliptic Curves

LMSST: 24 Lectures on Elliptic Curves
Author: John William Scott Cassels
Publisher: Cambridge University Press
Total Pages: 148
Release: 1991-11-21
Genre: Mathematics
ISBN: 9780521425308

A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.

LMSST

LMSST
Author: J. W. S. Cassels
Publisher:
Total Pages: 146
Release: 1991
Genre: Curves, Elliptic
ISBN: 9781107094505

A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.

Rational Points on Elliptic Curves

Rational Points on Elliptic Curves
Author: Joseph H. Silverman
Publisher: Springer Science & Business Media
Total Pages: 292
Release: 2013-04-17
Genre: Mathematics
ISBN: 1475742525

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.

Arithmetic Theory of Elliptic Curves

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.

Elliptic Curves (Second Edition)

Elliptic Curves (Second Edition)
Author: James S Milne
Publisher: World Scientific
Total Pages: 319
Release: 2020-08-20
Genre: Mathematics
ISBN: 9811221855

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.

Higher Regulators, Algebraic $K$-Theory, and Zeta Functions of Elliptic Curves

Higher Regulators, Algebraic $K$-Theory, and Zeta Functions of Elliptic Curves
Author: Spencer J. Bloch
Publisher: American Mathematical Soc.
Total Pages: 114
Release: 2011
Genre: Mathematics
ISBN: 0821829734

This is the long-awaited publication of the famous Irvine lectures. Delivered in 1978 at the University of California at Irvine, these lectures turned out to be an entry point to several intimately-connected new branches of arithmetic algebraic geometry, such as regulators and special values of L-functions of algebraic varieties, explicit formulas for them in terms of polylogarithms, the theory of algebraic cycles, and eventually the general theory of mixed motives which unifies and underlies all of the above (and much more).

Abelian l-Adic Representations and Elliptic Curves

Abelian l-Adic Representations and Elliptic Curves
Author: Jean-Pierre Serre
Publisher: CRC Press
Total Pages: 203
Release: 1997-11-15
Genre: Mathematics
ISBN: 1439863865

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Lectures on the Mordell-Weil Theorem

Lectures on the Mordell-Weil Theorem
Author: Jean-P. Serre
Publisher: Springer Science & Business Media
Total Pages: 228
Release: 2013-06-29
Genre: Technology & Engineering
ISBN: 3663106322

The book is based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve. Included are applications to, for example, Mordell's conjecture, the construction of Galois extensions, and the classical class number 1 problem. Comprehensive bibliographical references.

The Arithmetic of Elliptic Curves

The Arithmetic of Elliptic Curves
Author: Joseph H. Silverman
Publisher: Springer Science & Business Media
Total Pages: 414
Release: 2013-03-09
Genre: Mathematics
ISBN: 1475719205

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.

Elliptic Curves, Modular Forms, and Their L-functions

Elliptic Curves, Modular Forms, and Their L-functions
Author: Álvaro Lozano-Robledo
Publisher: American Mathematical Soc.
Total Pages: 217
Release: 2011
Genre: Mathematics
ISBN: 0821852426

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.