Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^{\times }$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1\mathrm{/}2,f\times {\theta }_{\chi })$, where $f$ is a (varying) newform on $D^{\times }$ of level $p^n$, and ${\theta }_{\chi }$ is an (essentially fixed) automorphic form on ${\mathrm{G}\mathrm{L}}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to \emph{any} family whose associated matrix coefficients have such a decay property.