Elliptic Curves, Modular Forms, and Their L-functions
Title | Elliptic Curves, Modular Forms, and Their L-functions PDF eBook |
Author | Álvaro Lozano-Robledo |
Publisher | American Mathematical Soc. |
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.
Introduction to Elliptic Curves and Modular Forms
Title | Introduction to Elliptic Curves and Modular Forms PDF eBook |
Author | Neal I. Koblitz |
Publisher | Springer Science & Business Media |
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.
Elliptic Curves
Title | Elliptic Curves PDF eBook |
Author | Henry McKean |
Publisher | Cambridge University Press |
Pages | 300 |
Release | 1999-08-13 |
Genre | Mathematics |
ISBN | 9780521658171 |
An introductory 1997 account in the style of the original discoverers, treating the fundamental themes even-handedly.
The 1-2-3 of Modular Forms
Title | The 1-2-3 of Modular Forms PDF eBook |
Author | Jan Hendrik Bruinier |
Publisher | Springer Science & Business Media |
Pages | 273 |
Release | 2008-02-10 |
Genre | Mathematics |
ISBN | 3540741194 |
This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.
Heads in Grammatical Theory
Title | Heads in Grammatical Theory PDF eBook |
Author | Greville G. Corbett |
Publisher | Cambridge University Press |
Pages | 364 |
Release | 1993-06-24 |
Genre | Language Arts & Disciplines |
ISBN | 9780521402453 |
A study of the idea of the 'head' or dominating element of a phrase.
Modular Forms and Fermat’s Last Theorem
Title | Modular Forms and Fermat’s Last Theorem PDF eBook |
Author | Gary Cornell |
Publisher | Springer Science & Business Media |
Pages | 592 |
Release | 2013-12-01 |
Genre | Mathematics |
ISBN | 1461219744 |
This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
Elliptic Curves, Modular Forms and Cryptography
Title | Elliptic Curves, Modular Forms and Cryptography PDF eBook |
Author | Ashwani K. Bhandari |
Publisher | Springer |
Pages | 339 |
Release | 2003-07-15 |
Genre | Mathematics |
ISBN | 9386279150 |