Abstract
We study the $p$-adic analogue of the arithmetic Gan–Gross–Prasad (GGP) conjectures for unitary groups. Let $\Pi$ be a conjugate-selfdual cuspidal automorphic representation of $\mathrm{GL}_n \times \mathrm{GL}_{n+1}$ over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of $L$-values of $\Pi$ appearing in the GGP conjectures. Then, when $\Pi$ is $p$-ordinary at a prime $p$, we construct a cyclotomic $p$-adic $L$-function $\mathscr{L}_p(\mathrm{M}_\Pi)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $\mathscr{L}_p(\mathrm{M}_\Pi)$ to the $p$-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the $p$-adic Beilinson–Bloch–Kato conjecture for the motive attached to $\Pi$. All proofs are based on some relative-trace formula in $p$-adic coefficients.
