Subconvexity for $L$-functions on ${\rm U}(n) \times {\rm U}(n+1)$ in the depth aspect

Abstract

Let $E/F$ be a CM extension of number fields, and let $H < G$ be a unitary Gan--Gross--Prasad pair defined with respect to $E/F$ that is compact at infinity. We consider a family $\mathcal{F}$ of automorphic representations of $G \times H$ that is varying at a finite place $w$ that splits in $E/F$. We assume that the representations in $\mathcal{F}$ satisfy certain conditions, including being tempered and distinguished by the GGP period. For a representation $\pi \times \pi_H \in \mathcal{F}$ with base change $\Pi \times \Pi_H$ to ${\rm GL}_{n+1}(E) \times {\rm GL}_n(E)$, we prove a subconvex bound \[ L(1/2, \Pi \times \Pi_H^\vee) \ll C(\Pi \times \Pi_H^\vee)^{1/4 - \delta} \] for any $\delta < \tfrac{1}{4n(n+1)(2n^2 + 3n + 3)}$. Our proof uses the unitary Ichino--Ikeda period formula to relate the central $L$-value to an automorphic period, before bounding that period using the amplification method of Iwaniec--Sarnak.