The Birch and Swinnerton-Dyer (BSD) conjecture relates the arithmetic rank of a rational elliptic curve to the order of vanishing of its L-function. 

After the contributions of Coates, Wiles and Rubin to the rank-zero case, and Gross, Zagier and Kolyvagin to rank one, many different approaches have been developed. The approaches address the higher-rank case, a p-adic reformulation of the conjecture, and a generalization of the conjecture to the framework of Galois representations. The latter is known as the Galois-equivariant BSD conjecture. Works pioneered by Mazur, Tate, Teitelbaum, Kato and others have rephrased and substantially developed the p-adic analogue of the conjecture. On the other hand, Skinner, Darmon, Bertolini, Rotger, Fornea, Sols, Blanco-Chacón and others have addressed the Galois-equivariant conjecture and the higher-rank case.

Despite all the efforts to establish the classical conjecture beyond rank one, there is still no effective clue on how to relate the second-order derivative of the L-function to global algebraic points, which take the role that Heegner points play in the rank-one case.

It is then worth looking back and examining the seminal first results on the BSD conjecture. 

Rephrasing (and bastardizing) Antonio Machado's poem:

The goal of this summer school is to introduce the theory of Heegner points and the celebrated Gross–Zagier formula, which were used to prove the rank-one case of the BSD conjecture. We will start by providing some basics on ray class fields, complex multiplication, and the Artin and Shimura reciprocity laws. After that, we will present the main properties of Heegner systems and the Heegner hypothesis, including some computational aspects carried out with Magma. We will continue with an overview of the proof of Gross–Zagier–Kolyvagin's Theorem, using Heegner systems and earlier ideas by Waldspurger. Finally, we will study Gross–Zagier's 1986 article: first, we will relate the derivative of the L-function to the Petersson inner product of certain newforms; second, we will express this product as the Néron–Tate pairing at a Heegner point. Every afternoon, there will be a discussion period to clear any doubts and freely discuss ideas on the subject.

The summer school is intended for MSc students and PhD students. The ideal prerequisites are a course on algebraic number theory and some basics on elliptic curves.