# Analytic torsion for surfaces with cusps I. Compact perturbation theorem and anomaly formula

## TL;DR

This paper develops a theory of analytic torsion for punctured Riemann surfaces with cusps, establishing a perturbation theorem, anomaly formula, and compatibility with Selberg trace formula, advancing understanding of degenerating hyperbolic surfaces.

## Contribution

It introduces a new definition of analytic torsion for surfaces with cusps, proves a compact perturbation theorem and anomaly formula, and shows compatibility with existing Selberg trace formula approaches.

## Key findings

- Analytic torsion relates to non-cusped surfaces' torsion.
- Established anomaly formula for Quillen norm.
- Proved compatibility with Takhtajan-Zograf torsion via Selberg trace formula.

## Abstract

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincar\'e metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.