Softcover ISBN:  9780821852422 
Product Code:  STML/58 
List Price:  $42.00 
Individual Price:  $33.60 
Electronic ISBN:  9781470416416 
Product Code:  STML/58.E 
List Price:  $39.00 
Individual Price:  $31.20 

Book DetailsStudent Mathematical LibraryIAS/Park City Mathematics SubseriesVolume: 58; 2011; 195 ppMSC: Primary 14; 11;
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.
LozanoRobledo 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, LozanoRobledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and SwinnertonDyer 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.This book is published in cooperation with IAS/Park City Mathematics Institute.ReadershipUndergraduate and graduate students interested in number theory and \(L\)functions.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Elliptic curves

Chapter 3. Modular curves

Chapter 4. Modular forms

Chapter 5. $L$functions

Appendix A. PARI/GP and Sage

Appendix B. Complex analysis

Appendix C. Projective space

Appendix D. The $p$adic numbers

Appendix E. Parametrization of torsion structures


Additional Material

Reviews

...ambitious undergraduates need a chance to get to know the fabled cities, and now they have it. With any luck, they will fall in love with them and come help us explore their mysteries.
MAA Reviews 
A welcome addition to a serious mathematics library.
CHOICE Reviews 
I think this is a wonderful book. In less than 200 pages LozanoRobledo covers a solid amount of modern number theory in a manner altogether accessible to a novice, and in a fashion so as to convey number theory's irresistible beauty. . . . [T]he themes LozanoRobledo addresses . . . are deep and sophisticated; considerable mathematical background is required for their mastery. But mastery is not what LozanoRobledo is after: it's more along the lines of rendering 'the primary objects of study, the statements of the main theorems, and their corollaries . . . within the grasp of advanced undergraduates.' And in this objective LozanoRobledo succeeds admirably. The book is full of examples and exercises of such appeal that a properly disposed rookie should go after nighon all of them; to boot, the author's narrative is compact and smooth. Elliptic Curves, Modular Forms, and Their LFunctions is a marvelous addition to the literature. Had I had it available as a kid, it would have been among my very favorites!
Michael Berg, MAA Reviews 
The most remarkable aspect [of the book] is the emphasis on detailed analysis of the definitions and complete explanation of the statements of the main theorems and corollaries.
J. R. Delgado, European Mathematical Society


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 Book Details
 Table of Contents
 Additional Material
 Reviews

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 Get Permissions
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.
LozanoRobledo 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, LozanoRobledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and SwinnertonDyer 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.
Undergraduate and graduate students interested in number theory and \(L\)functions.

Chapters

Chapter 1. Introduction

Chapter 2. Elliptic curves

Chapter 3. Modular curves

Chapter 4. Modular forms

Chapter 5. $L$functions

Appendix A. PARI/GP and Sage

Appendix B. Complex analysis

Appendix C. Projective space

Appendix D. The $p$adic numbers

Appendix E. Parametrization of torsion structures

...ambitious undergraduates need a chance to get to know the fabled cities, and now they have it. With any luck, they will fall in love with them and come help us explore their mysteries.
MAA Reviews 
A welcome addition to a serious mathematics library.
CHOICE Reviews 
I think this is a wonderful book. In less than 200 pages LozanoRobledo covers a solid amount of modern number theory in a manner altogether accessible to a novice, and in a fashion so as to convey number theory's irresistible beauty. . . . [T]he themes LozanoRobledo addresses . . . are deep and sophisticated; considerable mathematical background is required for their mastery. But mastery is not what LozanoRobledo is after: it's more along the lines of rendering 'the primary objects of study, the statements of the main theorems, and their corollaries . . . within the grasp of advanced undergraduates.' And in this objective LozanoRobledo succeeds admirably. The book is full of examples and exercises of such appeal that a properly disposed rookie should go after nighon all of them; to boot, the author's narrative is compact and smooth. Elliptic Curves, Modular Forms, and Their LFunctions is a marvelous addition to the literature. Had I had it available as a kid, it would have been among my very favorites!
Michael Berg, MAA Reviews 
The most remarkable aspect [of the book] is the emphasis on detailed analysis of the definitions and complete explanation of the statements of the main theorems and corollaries.
J. R. Delgado, European Mathematical Society