Beilinson-Kato elements and rational points on elliptic curves

We discuss Iwasawa theory and $p$-adic analogues of the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves defined over the rationals, focusing on the 'exceptional-zero' situation according to Mazur-Tate-Teitelbaum. After a general introduction, we will outline a strategy aimed at proving the $p$-adic BSD conjecture for elliptic curves of rank one, and grounded on Hida theory of p-adic families of modular forms. As a crucial step in the strategy, we relate Beilinson-Kato elements to Heegner points, establishing a conjecture of Perrin-Riou.