The p-adic Gross-Zagier formula on Shimura curves

We prove a general formula for the -adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above . The formula is in terms of the cyclotomic derivative of a Rankin-Selberg -adic -function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is rather than , by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.