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.
Readership
Undergraduate and graduate students interested in
number theory and $L$-functions.
