The goal of the mini course shall be to discuss the Birch and Swinnerton Dyer (BSD) conjecture. One of the most celebrated open problems in number theory, it is among the Clay Institute's seven "millenium prize problems". Attempts to better understand and generalise it form the basis for some of the most fundamental advances in the subject over the last five decades.

Since its formulation in the early 60's, progress on the BSD conjecture has come principally from the following directions:

1. The seminal work of Coates and Wiles in the late 70's which established the very first cases of the conjecture, for elliptic curves with complex multiplication and analytic rank 0.

2. The combined breakthrough by Gross-Zagier and Kolyvagin in the 1980's, which leads to a proof of the BSD conjecture for all (modular) elliptic curves over

3. The work of Kato in the 1990's establishing the BSD conjecture in analytic rank zero, for (modular) elliptic curves over

4. The more recent complementary approach of Skinner and Urban from the last decade which, inspired from earlier work of Mazur and Wiles on the arithmetic of cyclotomic fields, has led to a proof of the Birch and Swinnerton-Dyer conjecture for all modular elliptic curves of

After presenting a broad outline of these approaches, with emphasis on their fundamental similarities and differences, the course will be devoted to recent progress on the Birch and Swinnerton-Dyer conjecture for Mordell-Weil groups over ring class fields of real quadratic fields. Eventual connections with the theory of Stark-Heegner points and the problem of "explicit class field theory" for real quadratic fields will be addressed.

The course will assume basic background in the arithmetic of elliptic curves, such as might be covered in volume I of Silverman's book, as well as some background on modular forms and their connection with elliptic curves via the Shimura Taniyama conjecture. An excellent reference for this material is the textbook by Diamond and Shurman. A terse synopsis of the required material can also be found in the first three chapters of the book

Henri Darmon. Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, 101. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.

Chapter 10 also cover more specialised, but very relevant, material on the method of Kolyvagin, and Chapters 8 and 9 give a brief and somewhat outdated account of the theory of Stark-Heegner points. Better references for these two topics are, respectively, Gross's article on Kolyvagin's method which appeared in the Proceedings of the Durham Symposium in 1989, and the article of Matthew Greenberg that appears in the Duke Math Journal (vol 147) in 2009.

Henri Darmon. Heegner points, Stark-Heegner points, and values of L-series. International Congress of Mathematicians. Vol. II, 313-345, Eur. Math. Soc., Zurich, 2006.

Henri Darmon. Cycles on modular varieties and rational points on elliptic curves. Oberwolfach reports, Volume 6, Issue 3, 2009, 1843-1920 Explicit Methods in Number Theory.

Massimo Bertolini, Francesc Castella, Henri Darmon, Samit Dasgupta, Kartik Prasanna, and Victor Rotger.

Henri Darmon and Victor Rotger. Diagonal cycles and Euler systems I: A p-adic Gross-Zagier formula. Annales Scientifiques de l'Ecole Normale Supérieure, to appear.

Henri Darmon and Victor Rotger. Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series. Submitted.

Henri Darmon, Victor Rotger, and Yu Zhao. The Birch and Swinnerton-Dyer conjecture for Q-curves and Oda's period relations. ``Geometry and Analysis of Automorphic forms of several variables", Proceedings of the International Symposium in Honor of Takayuki Oda on the Occasion of his 60th birthday, Series on Number Theory and its applications, Vol. 7, Yoshinori Hamahata, Takashi Ichikawa, Atsushi Murase, and Takashi Sugano, eds., World Scientific, 2012, pp. 1-40.