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Resolvent Trace Formula and Determinants of $\boldsymbol{n}$ Laplacians on Orbifold Riemann Surfaces
Lee-Peng Teo Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, 43900, Sepang, Selangor, Malaysia
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 $\det\!'\Delta_n$. 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.
Keywords:
determinant of Laplacian, $n$-differentials, cocompact Riemann surfaces, Selberg trace formula.
Received: April 7, 2021; in final form September 5, 2021; Published online September 13, 2021
Citation:
Lee-Peng Teo, “Resolvent Trace Formula and Determinants of $\boldsymbol{n}$ Laplacians on Orbifold Riemann Surfaces”, SIGMA, 17 (2021), 083, 40 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1765 https://www.mathnet.ru/eng/sigma/v17/p83
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Abstract page: | 60 | Full-text PDF : | 18 | References: | 13 |
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