By Álvaro Lozano-Robledo

Many difficulties in quantity concept have uncomplicated statements, yet their suggestions require a deep figuring out of algebra, algebraic geometry, complicated research, workforce representations, or a mixture of all 4. the unique easily acknowledged challenge will be obscured within the intensity of the speculation built to appreciate it. This publication is an creation to a couple of those difficulties, and an outline of the theories used these days to assault them, offered in order that the quantity concept is usually on the vanguard of the dialogue. Lozano-Robledo offers an introductory survey of elliptic curves, modular types, and $L$-functions. His major aim is to supply the reader with the large photograph of the magnificent connections between those 3 households of mathematical items and their which means for quantity conception. As a for instance, Lozano-Robledo explains the modularity theorem and its recognized end result, Fermat's final Theorem. He additionally discusses the Birch and Swinnerton-Dyer Conjecture and different sleek conjectures. The publication starts with a few motivating difficulties and comprises a variety of concrete examples in the course of the textual content, frequently regarding real numbers, akin to three, four, five, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular types, and $L$-functions are too big to be coated in one quantity, and their proofs are outdoor the scope of the undergraduate curriculum. in spite of the fact that, the first gadgets of analysis, the statements of the most theorems, and their corollaries are in the snatch of complex undergraduates. This ebook concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and delivering plenty of examples, instead of living at the tough proofs. The booklet succeeds if, after examining the textual content, scholars consider pressured to review elliptic curves and modular varieties in all their glory