One Semester of Elliptic Curves
This rather short book gives an introduction to all important aspects of the theory of elliptic curves. As a consequence, many numerically or algebraically demanding calculations are left to the reader, often in the form of exercises. The book starts with a discussion of some analytic aspects of elliptic curves (including elliptic integrals, elliptic functions and projective realizations of elliptic curves). A more algebraic approach is used in a description of the standard correspondence between equivalence classes of elliptic curves and lattices, thereby leading to the j-function. By the end, the author has turned to more advanced topics like counting points on elliptic curves, curves with complex multiplication and the use of modular forms for proving the Jacobi formula for the number of representations of a positive integer as a sum of four squares. The book will be useful both for students of mathematics and computer science.