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

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 à 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 $\det {}^{\prime }{\Delta }_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det {}^{\prime }{\Delta }_n$ is equal to a constant ${\mathcal{C}}_n$ times $Z(n)$ when $n\ge 2$. 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.

References in corpus (1)

• Riemann-Roch isometries in the non-compact orbifold setting