Synopsis
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.
Review
...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
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
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