The refined arithmetic Gan–Gross–Prasad conjecture for p-adic heights

Let $V\in {\mathscr{V}}^{\circ}$ be an incoherent pair, and let $\pi$ be a distinguished, stable, ordinary, cuspidal automorphic representation of $\mathrm{G}^{V}(\mathbf{A})$, trivial at infinity, over a finite extension $L$ of $\mathbf{Q}_{p}$. Let $\Pi\coloneqq {\rm BC}(\pi)$, and assume that it is ordinary and non-exceptional. For $\phi\in\pi$ and $\phi'\in\pi^{\vee}$, let $h_{\pi}$ be the $p$-adic height pairing, $Z_{\pi}$ and $Z_{\pi^{\vee}}$ the associated Gan–Gross–Prasad cycles, $e_{p}({\mathrm {M}}\xspace_{\Pi})$ the local $p$-adic factor, $\partial {\mathscr{L}}\xspace_{p}({\mathrm {M}}\xspace_{\Pi})$ the derivative of the $p$-adic $L$-function, and $\alpha$ the canonical invariant functional. The refined arithmetic Gan–Gross–Prasad conjecture. For all $\phi\in\pi$ and $\phi'\in\pi^{\vee}$, one has

$$h_{\pi}(Z_{\pi}(\phi),Z_{\pi^{\vee}}(\phi'))=e_{p}({\mathrm {M}}\xspace_{\Pi})^{-1}\cdot {1\over 4}\partial {\mathscr{L}}\xspace_{p}({\mathrm {M}}\xspace_{\Pi})\cdot\alpha(\phi,\phi')$$

in $\Gamma_{F_{0}}\hat{\otimes}L$. This is presented as a $p$-adic analogue of the refined arithmetic Gan–Gross–Prasad conjecture and is related, when $n=1$, to the $p$-adic Gross–Zagier formula.