Many difficulties in quantity conception have basic statements, yet their suggestions require a deep knowing of algebra, algebraic geometry, advanced research, crew representations, or a mix of all 4. the unique easily acknowledged challenge will be obscured within the intensity of the idea built to appreciate it. This e-book is an creation to a couple of those difficulties, and an summary of the theories used these days to assault them, awarded in order that the quantity conception 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 massive photograph of the fantastic connections between those 3 households of mathematical gadgets and their that means for quantity thought. As a for instance, Lozano-Robledo explains the modularity theorem and its well-known end result, Fermat's final Theorem. He additionally discusses the Birch and Swinnerton-Dyer Conjecture and different smooth conjectures. The e-book starts with a few motivating difficulties and comprises a number of concrete examples in the course of the textual content, frequently related to real numbers, similar to three, four, five, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular types, and $L$-functions are too giant to be coated in one quantity, and their proofs are outdoor the scope of the undergraduate curriculum. even if, the first gadgets of research, the statements of the most theorems, and their corollaries are in the seize of complex undergraduates. This publication concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and delivering plenty of examples, instead of residing at the tough proofs. The booklet succeeds if, after studying the textual content, scholars think forced to check elliptic curves and modular kinds in all their glory.

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