Elliptic Curves, Modular Forms, and Their L-functions
The title of this book clearly explains its content: the author, Alvaro Lozano-Robledo, gives an introduction to the study of elliptic curves, modular forms and L-functions. The standard reader of the text is an undergraduate student; in fact, the origin of the book is a course that the author taught at an undergraduate summer school in 2009. The text presents the relationships between the curves, forms and functions mentioned. The most remarkable aspect is the emphasis on detailed analysis of the definitions and complete explanation of the statements of the main theorems and corollaries. With respect to demonstrations, the simplest are left to the reader and, for the most important theorems, the author gives references. The book consists of five chapters whose contents are as follows: In Chapter 1 the author motivates the principal concepts using some classical problems (congruent numbers, FLT, representation of integers); Chapter 2 -the longest in the book- is a brief report on the arithmetic of elliptic curves (integer points, group of rational points, torsion and rank,...) which includes a proof of the weak theorem of Mordell-Weil using the method of 2-descent; Chapter 3 discusses various relationships between elliptic curves and modular curves and Chapter 4 discusses modular forms for the modular group and other subgroups of congruences; finally, in Chapter 5 the author defines the L-functions associated with modular curves, and briefly introduces the Birch and Swinnerton-Dyer conjecture and the Shimura-Taniyama-Weil conjecture. The book is supplemented by five appendices (including a reference to the software packages PARI and Sage) and four pages of references (it is remarkable that JWS Cassels is not cited at all).