We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo–Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky–Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change.

中文翻译：

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二次赫克和和质量均分

我们考虑紧算术双曲曲面上全纯 Hecke 特征形的量子唯一遍历性猜想的类比。我们表明，这个猜想来自对二次级数求和的 Hecke 特征值的非平凡界限。我们的归约为罗-萨纳克为非紧凑模块化表面的情况建立的标准的紧凑情况提供了一个类比。新颖之处在于，此类标准的已知证明主要依赖于傅里叶展开，而这在紧凑情况下是不可用的。无条件地，我们建立了 Holowinsky-Soundararajan 定理的扭曲变体，涉及通过基数变化产生的归一化希尔伯特模形式的限制。