轨道折叠Riemann Surfaces上$n$ Laplacian的预解迹公式和行列式
Resolvent Trace Formula and Determinants of $n$ Laplacians on Orbifold Riemann Surfaces
摘要：
对于非负整数$n$，我们考虑作用在具有分歧点的有限Riemann曲面$X$上的$n$ -微分空间上的$n$ -Laplacian $Δ_n$.沿着La Selberg线发展了预解式核的迹公式。利用迹公式计算了$Δ_n+s(s+2n-1)$的正则化行列式，并由此导出了$Δ_n$的正则化行列式，记为$\det\!'Δ_n$ .考虑到来自绝对连续光谱的贡献，当$n\geq 2$时，$\det\!'Δ_n$等于常数$\mathcal{C}_n$乘以$Z(n)$。这里$Z(s)$是$X$的Selberg zeta函数。当$n=0$或$n=1$时，$Z(n)$被分别围绕$s=0$和$s=1$的$Z(s)$的泰勒展开式的前导系数替换。常数$\mathcal{C}_n$是明确计算的。它们依赖于亏格，尖点的数目，以及分歧指数，但与模量参数无关。
英文摘要：
For $n$ a nonnegative integer, we consider the $n$-Laplacian $Δ_n$ 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 $Δ_n+s(s+2n-1)$, from which we deduce the regularized determinant of $Δ_n$, denoted by $\det\!'Δ_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det\!'Δ_n$ 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.
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