# Publication list for Lozano-Robledo, Álvaro

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Title (HTML): Number Theory and Geometry: An Introduction to Arithmetic Geometry
Author(s) (Product display): Álvaro Lozano-Robledo
Affiliation(s) (HTML): University of Connecticut, Storrs, CT
Custom URL: amstext-35
Abstract:

Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. Similarly, Gauss's law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively.

This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.

Book Series Name: Pure and Applied Undergraduate Texts
Volume: 35
Publication Month and Year: 2019-03-21
Page Count: 488
Cover Type: Hardcover
Print ISBN-13: 978-1-4704-5016-8
Online ISBN 13: 978-1-4704-5190-5
Print ISSN: 1943-9334
Online ISSN: 1943-9334
Primary MSC: 11; 14
Textbook?: true
Applied Math?: false
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Electronic Media?: false
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SXG Subject: NT
Supplemental Materials (URL at AMS): https://www.ams.org/bookpages/amstext-35
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Online Price 1: 109.00
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Exam/Desk Copy: https://www.ams.org/exam-desk-review-request?&eisbn=978-1-4704-5190-5&pisbn=978-1-4704-5016-8&epc=AMSTEXT/35.E&ppc=AMSTEXT/35&title=Number%20Theory%20and%20Geometry%3A%20An%20Introduction%20to%20Arithmetic%20Geometry&author=%C1lvaro%20Lozano-Robledo&type=DE
Review Copy: https://www.ams.org/exam-desk-review-request?&eisbn=978-1-4704-5190-5&pisbn=978-1-4704-5016-8&epc=AMSTEXT/35.E&ppc=AMSTEXT/35&title=Number%20Theory%20and%20Geometry%3A%20An%20Introduction%20to%20Arithmetic%20Geometry&author=%C1lvaro%20Lozano-Robledo&type=R

Reviews:

I think this book would be a great foundation for a course which is more inspiring—and perhaps more challenging—than your standard course on elementary number theory.

-- Abbey Bourdon, MAA Reviews

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• Base Product Code Keyword List: stml; STML; stml/58; STML/58; stml-58; STML-58
Print Product Code: STML/58
Online Product Code: STML/58.E
Title (HTML): Elliptic Curves, Modular Forms, and Their L-functions
Author(s) (Product display): Álvaro Lozano-Robledo
Affiliation(s) (HTML): University of Connecticut, Storrs, CT
Custom URL: stml-58
Abstract:

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

Book Series Name: Student Mathematical Library
Volume: 58
Publication Month and Year: 2011-02-08
Page Count: 195
Cover Type: Softcover
Print ISBN-13: 978-0-8218-5242-2
Online ISBN 13: 978-1-4704-1641-6
Print ISSN: 1520-9121
Online ISSN: 1520-9121
Primary MSC: 14; 11
Textbook?: false
Applied Math?: false
MAA Book?: false
Electronic Media?: false
Publisher (non-AMS): Published in cooperation with IAS/Park City Mathematics Institute
SXG Subject: NT
Supplemental Materials (URL at AMS): https://www.ams.org/bookpages/stml-58
Online Price 1 Label: List
Online Price 1: 39.00
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Print Price 1: 42.00
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Undergraduate and graduate students interested in number theory and $$L$$-functions.

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.

Michael Berg, MAA 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!

-- CHOICE 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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