Preface This book grew out of the lecture notes for a course on “Elliptic Curves, Modular Forms and L-functions” that the author taught at an undergraduate summer school as part of the 2009 Park City Mathe- matics Institute. These notes are an introductory survey of the theory of elliptic curves, modular forms and their L-functions, with an em- phasis on examples rather than proofs. The main goal is to provide the reader with a big picture of the surprising connections among these three types of mathematical objects, which are seemingly so distinct. In that vein, one of the themes of the book is to explain the statement of the modularity theorem (Theorem 5.4.6), previously known as the Taniyama-Shimura-Weil conjecture (Conjecture 5.4.5). In order to underscore the importance of the modularity theorem, we also discuss in some detail one of its most renowned consequences: Fermat’s last theorem (Example 1.1.5 and Section 5.5). It would be impossible to give the proofs of the main theorems on elliptic curves and modular forms in one single course, and the proofs would be outside the scope of the undergraduate curriculum. However, the definitions, the statements of the main theorems and their corollaries can be easily understood by students with some stan- dard undergraduate background (calculus, linear algebra, elementary number theory and a first course in abstract algebra). Proofs that are accessible to a student are left to the reader and proposed as exercises xi
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