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