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 $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ 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 any family whose associated matrix coefficients have such a decay property.

中文翻译：

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紧算术曲面水平方面的本征函数的超范数，II：新形式和次凸性

我们改进了 $D^\times$ 上新形式的超范数的深度方面的局部边界，其中 $D$ 是 ${\mathbb {Q}}$ 上的不定四元数除法代数。我们的 sup-norm 边界意味着 $L(1/2, f \times \theta _\chi )$ 的深度-方面次凸边界，其中 $f$ 是 $D^\times$ 级别上的（变化的）新形式$p^n$ 和 $\theta _\chi$ 是 $\textrm {GL}_2$ 上的（基本上固定的）自守形式，作为 Hecke 字符 $\chi$ 在二次域上的 theta 提升获得。为了证明，我们用一个新的过滤参数和最近由第二名作者证明的计数结果来增强放大方法，以减少显示沿着紧凑子集的局部新向量的矩阵系数的强定量衰减，我们通过 $p$ 建立-adic 固定相分析。此外，