By Álvaro Lozano-Robledo

Many difficulties in quantity conception have basic statements, yet their suggestions require a deep knowing of algebra, algebraic geometry, advanced research, workforce representations, or a mix of all 4. the unique easily acknowledged challenge should be obscured within the intensity of the speculation constructed to appreciate it. This e-book is an advent to a few of those difficulties, and an summary of the theories used these days to assault them, offered in order that the quantity concept is often on the leading edge of the dialogue. Lozano-Robledo supplies an introductory survey of elliptic curves, modular types, and $L$-functions. His major objective is to supply the reader with the massive photo of the remarkable connections between those 3 households of mathematical gadgets and their which means for quantity thought. As a working example, Lozano-Robledo explains the modularity theorem and its recognized final result, Fermat's final Theorem. He additionally discusses the Birch and Swinnerton-Dyer Conjecture and different sleek conjectures. The booklet starts off with a few motivating difficulties and contains quite a few concrete examples through the textual content, frequently related to genuine numbers, similar to three, four, five, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular kinds, and $L$-functions are too gigantic to be coated in one quantity, and their proofs are open air the scope of the undergraduate curriculum. besides the fact that, the first gadgets of research, the statements of the most theorems, and their corollaries are in the take hold of of complex undergraduates. This e-book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and offering plenty of examples, instead of living at the challenging proofs. The publication succeeds if, after examining the textual content, scholars think forced to check elliptic curves and modular varieties in all their glory