Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions

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 GL_{n} x 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 $L_p(M_\Pi)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $L_p(M_\Pi)$ 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 formulas in p-adic coefficients.