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

Let $E\mathrm{/}F$ be a CM extension of number fields, and let $H<G$ be a unitary Gan–Gross–Prasad pair defined with respect to $E\mathrm{/}F$ 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 $E\mathrm{/}F$. 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 $\mathrm{\Pi }\times {\mathrm{\Pi }}_H$ to ${\mathrm{G}\mathrm{L}}_{n+1}(E)\times {\mathrm{G}\mathrm{L}}_n(E)$, we prove a subconvex bound $$L(1\mathrm{/}2,\mathrm{\Pi }\times {\mathrm{\Pi }}_H^{\vee })\ll C(\mathrm{\Pi }\times {\mathrm{\Pi }}_H^{\vee }{)}^{1\mathrm{/}4-\delta }$$ for any $\delta <\frac{1}{4n(n+1)(2n^2+3n+3)}$. 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.