Arithmetic of Elliptic Curves over $\mathbb{Q}$

The Birch and Swinnerton-Dyer conjecture relates the rank of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve $E/\mathbb{Q}$ to the order of vanishing of its $L$-function $L(E, s)$ at $s = 1$. We discuss two complementary lines of progress: Bhargava's work using geometry-of-numbers methods to bound average ranks (showing that a positive proportion of elliptic curves have rank $0$ and a positive proportion have rank $1$), and Zhang's Gross-Zagier type formula relating heights of Heegner points to derivatives of $L$-functions, establishing BSD \in analytic rank $1$ for many curves.