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

We obtain for the first time an improvement over the local bound in the depth aspect for sup-norms of newforms on an indefinite quaternion division algebra $D$ over $\mathbb{Q}$. A central role in our method is played by the decay of local matrix coefficients. More generally, we prove a strong upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have a proper decay along a suitable sequence of compact subsets. We observe that our sup-norm bound for newforms directly implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta_\chi)$, where $f$ is a newform on $D^\times$ of level $p^n$, and $\theta_\chi$ is an (essentially fixed) automorphic form on $\mathrm{GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field.