Density of Resonances for Covers of Schottky Surfaces

TL;DR

This paper studies how resonance counting functions for Schottky surfaces change under finite covers, providing bounds and improved estimates using transfer operators and thermodynamic formalism, refining previous results in the field.

Contribution

It introduces new bounds for resonance counting functions under coverings of Schottky surfaces, using transfer operators and thermodynamic formalism, and improves existing estimates for fractal Weyl laws.

Findings

Bounds for resonance counting functions under surface covers.

An improved fractal Weyl upper bound depending on covering degree.

Refined estimates for principal congruence covers in the level aspect.

Abstract

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 Guillop\'{e}--Zworski and Guillop\'{e}--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.