The conjecture of Birch and Swinnerton-Dyer

This essay starts by first explaining, for elliptic curves defined over Q, the statement of the conjecture of Birch and Swinnerton-Dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the essay represent an account, with detailed proofs, of results about the cases of both week and strong Birch and Swinnerton-Dyer conjecture from the wonderful article by John Coates, Yongxiong Li, Ye Tian and Shuai Zhai [1]. Recently, working on the congruent number curve E : y2 = x3 − x, Ye Tian introduced a new method of attack for the following general problem. Problem. Given an elliptic curve E defined over Q, we would like to find a large explicit infinite family of square free integers M, coprime with the conductor C(E), such that L(E(M),s) has a simple zero at s = 1. Tian [21], [22] succeeded in doing this for his particular choice of curve, and, inspired by his work, the authors carry out this full programme for the elliptic curve A : y2 + xy = x3 − x2 −2x−1 in [1]. Mysteriously, this required them to prove a weak form of the 2-part of the Birch and Swinnerton-Dyer conjecture for an infinite family of quadratic twists of A, which is described at the end of the third section of this essay. The last section combines results from all the previous ones to prove the highlight of this essay, an analogue of Tian’s result for the elliptic curve A : y2 + xy = x3 − x2 −2x−1, formulated in Theorem 47. The autors of [1] belive that there should be analogues of this theorem for every elliptic curve E defined over Q and it is an important problem to formulate them precisely and then to prove them.