Taussky-Todd Lecture

In this talk I will aim to explain how many of the big tools for studying the arithmetic of elliptic curves over the rationals (the Gross--Zagier formula, Heegner points, Kato's Euler system, the Euler system of Beilinson--Flach classes) are related. And that these relations yield refined and even new results in the case of rank one curves. These results include: a proof of Perrin-Riou's conjecture on Kato's classes, an improved upper bound on the order of the Tate--Shafarevich group, the p-part of the BSD formula in the case of analytic rank one, and more. This is a report on joint work with Ashay Burungale and Ye Tian.