Let E be an elliptic curve over the field of rational numbers and let L(E, s) be its L-function. The Birch and Swinnerton-Dyer formula predicts that the first non-zero coefficient of the Taylor expansion of L(E, s) around s = 1 encodes some important arithmetic invariants attached to E. Assume that L(E, s) has a simple zero at s = 1.
By the work of Wei Zhang we know that the p-part of the BSD formula holds for all primes p greater or equal to 5. Recently, Chao Li has obtained some interesting results for p = 2. In this talk we shall discuss a refinement of the 2-part of the conjecture introduced in the Gross-Kohnen-Zagier (1987) paper that seems resonant with some aspects of Chao Li's recent results. The talk is intended for a non-specialist audience and we will focus mostly on some explicit examples.