Resolvent Trace Formula and Determinants of $n$ Laplacians on Orbifold Riemann Surfaces

TL;DR

This paper develops a Selberg-style trace formula for the resolvent kernel of n-Laplacians on orbifold Riemann surfaces, enabling explicit computation of their regularized determinants and relating them to the Selberg zeta function.

Contribution

It introduces a trace formula for the resolvent kernel of n-Laplacians on orbifold Riemann surfaces and computes their determinants explicitly in terms of the Selberg zeta function.

Findings

Derived a trace formula for the resolvent kernel of n-Laplacians.

Computed the regularized determinants of n-Laplacians explicitly.

Connected determinants to the Selberg zeta function and provided explicit constants.

Abstract

For $n$ a nonnegative integer, we consider the $n$-Laplacian $\Delta_n$ acting on the space of $n$-differentials on a confinite Riemann surface $X$ which has ramification points. The trace formula for the resolvent kernel is developed along the line \`a la Selberg. Using the trace formula, we compute the regularized determinant of $\Delta_n+s(s+2n-1)$, from which we deduce the regularized determinant of $\Delta_n$, denoted by $\det\!'\Delta_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det\!'\Delta_n$ is equal to a constant $\mathcal{C}_n$ times $Z(n)$ when $n\geq 2$. Here $Z(s)$ is the Selberg zeta function of $X$. When $n=0$ or $n=1$, $Z(n)$ is replaced by the leading coefficient of the Taylor expansion of $Z(s)$ around $s=0$ and $s=1$ respectively. The constants $\mathcal{C}_n$ are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but is independent of the moduli parameters.