Title: On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curve I & II

Abstract: In these two talks we discuss the Birch and Swinnerton-Dyer formula modulo square of rational numbers for the quadratic twist family of a given elliptic curve. In particular we show the following: let E be a semistable elliptic curve with conductor N, whose analytic rank is at most one, then for any positive fundamental discriminant D that is relatively prime to N, such that the quadratic twist E^D again has analytic rank at most one, we have that the Birch and Swinnerton-Dyer formula modulo square of rational numbers holds for E if and only of it holds for E^D. Joint work with Alexander Barrios.