Many difficulties in quantity conception have uncomplicated statements, yet their ideas require a deep realizing of algebra, algebraic geometry, complicated research, team representations, or a mix of all 4. the unique easily said challenge might be obscured within the intensity of the speculation constructed to appreciate it. This booklet is an advent to a couple of those difficulties, and an summary of the theories used these days to assault them, provided in order that the quantity concept is usually on the leading edge of the dialogue. Lozano-Robledo provides an introductory survey of elliptic curves, modular varieties, and $L$-functions. His major objective is to supply the reader with the large photo of the incredible connections between those 3 households of mathematical gadgets and their which means for quantity thought. 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 glossy conjectures. The e-book starts off with a few motivating difficulties and comprises various concrete examples during the textual content, frequently concerning real numbers, equivalent to three, four, five, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular varieties, and $L$-functions are too large to be lined in one quantity, and their proofs are outdoors the scope of the undergraduate curriculum. in spite of the fact that, the first items of analysis, the statements of the most theorems, and their corollaries are in the clutch of complex undergraduates. This e-book 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 ebook succeeds if, after examining the textual content, scholars think pressured to review elliptic curves and modular kinds in all their glory.