The universal p-adic Gross--Zagier formula

Let $G$ be the group $(GL_{2}\times GU(1))/GL_{1}$ over a totally real field $F$, and let $X$ be a Hida family for $G$. Revisiting a construction of Howard and Fouquet, we construct an explicit section $P$ of a sheaf of Selmer groups over $X$. We show, answering a question of Howard, that $P$ is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of $X$. We prove that the $p$-adic height of $P$ is given by the cyclotomic derivative of a $p$-adic $L$-function, conditionally on the existence of the latter. This formula over $X$ (which is an identity of functionals on some universal ordinary automorphic representation) specialises at classical points to all the Gross--Zagier formulas for $G$ that may be expected from representation-theoretic considerations. Combined with work of Fouquet, the formula implies the $p$-adic analogue of the Beilinson--Bloch--Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in $2[F:Q]+1$ variables. Other applications include two different generic non-vanishing results for Heegner classes and $p$-adic heights.