肖特基表面覆盖层的共振密度
Density of Resonances for Covers of Schottky Surfaces
摘要：
我们研究了肖特基表面的共振计数函数在过渡到有限度的覆盖面时的行为。我们认为经典的共振计数函数要求在大的（和不断增长的）磁盘共振的数量集中在原点的$\mathbb{C}$，以及（分形）共振计数函数要求的共振的数量在框附近的临界指数的轴。对于前者的计数函数，我们提供了一个转移算子为基础的证明，可以选择的边界常数，使下过渡到覆盖的转换行为是作为外尔定律的情况下，表面的有限面积。对于后一种计数函数，我们根据周期测地线在覆盖面上的覆盖度和最小长度导出了一个界。这产生了改进的分形Weyl上界。在肖特基表面的设置，这些估计完善以前的结果由于Guillopé-Zworski和Guillopé-Lin-Zworski。当应用于主同余覆盖时，这些结果产生了新的估计的共振计数函数的水平方面，这是最近由Jakobson-Naud研究。本文中使用的技术是基于$L$ -函数（扭曲Selberg zeta函数）和扭曲转移算子的热力学形式。
英文摘要：
We investigate how bounds of resonance counting functions for Schottky surfaces behave under transitions to covering surfaces of finite degree. We consider the classical resonance counting function asking for the number of resonances in large (and growing) disks centered at the origin of $\mathbb{C}$, as well as the (fractal) resonance counting function asking for the number of resonances in boxes near the axis of the critical exponent. For the former counting function we provide a transfer-operator-based proof that bounding constants can be chosen such that the transformation behavior under transition to covers is as for the Weyl law in the case of surfaces of finite area. For the latter counting function we deduce a bound in terms of the covering degree and the minimal length of a periodic geodesic on the covering surface. This yields an improved fractal Weyl upper bound. In the setting of Schottky surfaces, these estimates refine previous results due to Guillopé--Zworski and Guillopé--Lin--Zworski. When applied to principal congruence covers, these results yield new estimates for the resonance counting functions in the level aspect, which have recently been investigated by Jakobson--Naud. The techniques used in this article are based on the thermodynamic formalism for $L$-functions (twisted Selberg zeta functions), and twisted transfer operators.
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