Course Book: The Arithmetic of Elliptic Curves by Joe Silverman is the standard reference in the subject and should be owned by every number theorist. It takes an approach that is well in-line with modern arithmetic geometry, which is not exactly how this course will proceed. You will find this available in the U de M bookstore
Alternatives: Rational Points on Elliptic Curves by Joe Silverman and John Tate is more elementary but covers many of the key introductory issues from as elementary a viewpoint as possible. We will use the approaches from this book in much of the first half of the semester.
Introduction to Elliptic Curves and Modular Forms by Neal Koblitz was an early book on precisely the subject of this course. It is a beautiful coherent account of many ideas leading up to some extraordinary developments with modular forms.
I am providing some notes entitled Rational and Integral points on curves.
Classtimes: Monday 10h30-12h30; Wednesday 10h30-12h30, in Andre-Aisenstadt 5183.
Class dates: From Sept 7 to Dec 7, 2011