**Student Mathematical Library**

Volume: 58;
2011;
195 pp;
Softcover

MSC: Primary 14; 11;

**Print ISBN: 978-0-8218-5242-2
Product Code: STML/58**

List Price: $42.00

Individual Price: $33.60

**Electronic ISBN: 978-1-4704-1641-6
Product Code: STML/58.E**

List Price: $39.00

Individual Price: $31.20

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#### Supplemental Materials

# Elliptic Curves, Modular Forms, and Their L-functions

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*Álvaro Lozano-Robledo*

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.

This book is published in cooperation with IAS/Park City Mathematics Institute

#### Readership

Undergraduate and graduate students interested in number theory and \(L\)-functions.

#### Reviews & Endorsements

...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 Lozano-Robledo 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 Lozano-Robledo addresses . . . are deep and sophisticated; considerable mathematical background is required for their mastery. But mastery is not what Lozano-Robledo 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 Lozano-Robledo succeeds admirably. The book is full of examples and exercises of such appeal that a properly disposed rookie should go after nigh-on all of them; to boot, the author's narrative is compact and smooth. Elliptic Curves, Modular Forms, and Their L-Functions 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

#### Table of Contents

# Table of Contents

## Elliptic Curves, Modular Forms, and Their L-functions

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Introduction 116 free
- Elliptic curves 1732 free
- Modular curves 7792
- Modular forms 99114
- 𝐿-functions 123138
- PARI/GP and Sage 147162
- Complex analysis 159174
- Projective space 171186
- The 𝑝-adic numbers 179194
- Parametrization of torsion structures 185200
- Bibliography 189204
- Index 193208 free
- Back Cover Back Cover1214