Gross-Zagier formula: why is it right?

In his eccentrically written paper in 1952, Heegner showed that the 1000 years old congruent number problem is soluble for all prime $p$ (or $2p$) congruent to 5, 7 (or 6) mod 8, by constructing some rational points on certain elliptic curves of infinite order. In 1986, Gross and Zagier proved a formula which relates the heights of Heegner points and derivatives of $L$-series. At the ICM in 1986, Faltings asked: "Alles in allem handelt es sich um eine schöne Entdeckung, welche wir aber leider noch nicht "erklären" können: Warum ist sie richtig?"

In this talk, I will present a partial answer to Faltings' question. More precisely, I will present a representation theoretical framework (initiated by Gross) in which the Gross-Zagier formula and Waldspurger formula (on periods integrals of automorphic forms) can be viewed simultaneously. I will show how this framework be used to prove a general Gross-Zagier formula, and a general $p$-adic Waldspurger formula, and a general $p$-adic Gross-Zagier formula. If time permits, I will describe recent applications of Gross-Zagier formula to the congruent number problem, BSD conjecture, and the proposed generalization of Gross-Zagier formula to higher dimensional Shimura varieties.