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

We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let $Π$ 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 $Π$ appearing in the GGP conjectures. Then, when $Π$ is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function $L_p(M_Π)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $L_p(M_Π)$ 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 $Π$. All proofs are based on some relative-trace formulas in p-adic coefficients.

💡 Research Summary

The paper establishes a p‑adic analogue of the arithmetic Gan‑Gross‑Prasad (GGP) conjectures for unitary groups over a CM extension F/F₀. Starting with a conjugate‑selfdual cuspidal automorphic representation Π of GLₙ × GLₙ₊₁ that is algebraic of minimal regular weight (“trivial‑weight”), the authors first prove a strong rationality result (Theorem A): for every Hecke character χ of F, the twisted central value L(½, Π⊗χ) multiplied by an explicit epsilon‑factor belongs to the ring of algebraic integers. This generalises earlier work of Shimura, Grobner‑Lin and Li‑Liu‑Sun.

Assuming Π is p‑ordinary (i.e. admits a U‑eigenvector with unit eigenvalue at every p‑adic place), they construct a cyclotomic p‑adic L‑function Lₚ(M_Π) (Theorem B). The construction uses a p‑adic relative trace formula: a global automorphic distribution is matched with local p‑adic orbital integrals, and an explicit product of local factors eₚ(M_Π⊗χ) is identified. The resulting function interpolates the algebraic values from Theorem A and satisfies the expected interpolation formula.

The second half of the paper introduces Gan‑Gross‑Prasad cycles on unitary Shimura varieties (including the RSZ models) and defines their p‑adic heights via p‑adic Abel‑Jacobi maps and biextensions. A second p‑adic relative trace formula encodes these heights. By comparing the two trace formulas, the authors obtain a precise identity (Theorem D) relating the first derivative of Lₚ(M_Π) at the trivial character to the p‑adic height of the GGP cycle.

As an arithmetic application, they prove a p‑adic Beilinson‑Bloch‑Kato result (Theorem C). Under the sign condition ε(Π)=−1 and a collection of local hypotheses (unramifiedness at non‑split places, mild conductor bounds, and a non‑vanishing conjecture for certain local characters), the order of vanishing of Lₚ(M_Π) at the trivial character equals one, which forces the Bloch‑Kato Selmer group H¹_f(F, ρ_Π) to have dimension at least one. If, in addition, p is an admissible prime for Π (in the sense of recent Selmer‑bound work), the dimension is exactly one. This gives the first higher‑dimensional example where a p‑adic L‑function derivative controls a Selmer group, extending the classical Gross‑Zagier/Kolyvagin picture beyond the 2‑dimensional case.

The proofs combine several sophisticated tools: Jacquet–Rallis relative trace formulas, p‑adic orbital integral calculations, Gaussian test functions, Hasse measures, and the theory of integral models of Shimura varieties. The paper also outlines future directions, such as removing the unramified hypothesis at non‑split p‑adic places and extending the method to multi‑variable p‑adic families or other types of Shimura varieties. Overall, the work provides a complete and technically deep realization of the p‑adic Gan‑Gross‑Prasad program.