Seminar: Rational points on modular elliptic curves --------------------------------------------------- The goal of this seminar is to discuss the notion of Heegner points on modular elliptic curves as well as certain conjectural variants, and the information they provide on the Birch and Swinnerton-Dyer conjecture. Topic to be covered (roughly in chronological order) will include: 1. Classical Heegner points attached to imaginary quadratic fields. 2. Proof of Kolyvagin's theorem and its application to the Birch and Swinnerton-Dyer conjecture. 3. The rudiments of rigid analysis and p-adic uniformisation. 4. The notion of "Stark-Heegner" points introduced in [1]. 5. Relation with p-adic L-functions attached to Hida families. 6. A proof of some cases of the main conjecture of [1] (work in progress with M. Bertolini). References ---------- [1] H. Darmon. Integration on ${\cal H}_p\times{\cal H}$ and arithmetic applications. Annals of Mathematics 154 (2001) 589-639. [2] H. Darmon. Rational points on modular elliptic curves. NSF-CBMS Lectures, August 8-12 2001, to appear. Can be downloaded from http://www.math.mcgill.ca/darmon/pub/pub.html Note: it is better to dowload the .ps version if you want some figures to print correctly. [3] H. Darmon, Periods of modular forms and the arithmetic of elliptic curves, lectures at a summer school in Jussieu, July 2002. Can be downloaded (in streaming video) from http://www.institut.math.jussieu.fr/BSD/video.html Supplementary reading: ---------------------- Chapters I.1 and II.1 of C.L.Siegel. Advanced analytic number theory. This material covers Kronecker's classic solution to Pell's equation in terms of the Dedekind $\eta$-function. An understanding of this argument will be helpful when we get to topic 6, where the discussion proceeds along very similar lines.