AI 中文总结
该研究针对亏格趋于无穷的紧致双曲黎曼曲面序列，建立了加权素测地线定理的类似结果，并证明加权拉普拉斯算子谱行列式的对数与曲面体积的比值收敛到仅依赖于权极限的常数，结果与三类概率模型兼容。
AI 中文摘要
设$(X,\boldsymbol{\textit{\textchi}},k)$为三元组，其中$X$是亏格为$g$的光滑紧致双曲黎曼曲面，$\boldsymbol{\textit{\textchi}}$是维数为$m$的容许权酉乘子系统。第一项成果建立了对应于$(X,\boldsymbol{\textit{\textchi}},k)$的加权素测地线计数函数的素测地线定理的类似结果，所得误差项是显式的，其可有效计算的常数仅依赖于$X$的亏格、$\boldsymbol{\textit{\textchi}}$的维数、$X$上最短测地线的长度，以及加权拉普拉斯算子$\boldsymbol{\textit{\textDelta}}_{2k}$和标量拉普拉斯算子$\boldsymbol{\textit{\textDelta}}_{0}$的最小非零特征值。第二项成果研究了序列$(X_{n}, \boldsymbol{\textit{\textchi}}_{n}, k_{n})$的谱行列式$\boldsymbol{\textit{\textdet}}\boldsymbol{\textit{\textDelta}}_{2k_n}$的渐近行为，其中$X_n$的亏格趋于无穷。在相当一般的条件下，即存在弱谱间隙、基础富克斯群的一致离散性，以及有界测地线的某种非积累性，我们证明$\boldsymbol{\textit{\textlog}}\boldsymbol{\textit{\textdet}}\boldsymbol{\textit{\textDelta}}_{2k_n}/\boldsymbol{\textit{\textvol}}(X_{n})$收敛到仅依赖于$\boldsymbol{\textit{\textalpha}}=\boldsymbol{\textit{\textlim}}_{n\to\boldsymbol{\textit{\textinfty}}} k_n$的常数$C_{\boldsymbol{\textit{\textalpha}}}$。该结果是确定性的，且与三个已被深入研究的概率模型兼容，即Weil-Petersson模型、Brooks-Makover模型和随机覆盖模型。
英文摘要
Let $(X,χ,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $χ$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,χ,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $χ$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian $Δ_{2k}$ as well that of the scalar Laplacian $Δ_{0}$. Our second result studies the asymptotic behavior of the spectral determinant $\detΔ_{2k_n}$ for a sequence $(X_{n}, χ_{n}, k_{n})$ for which the genus of $X_n$ tends to infinity. Under reasonably general circumstances, namely the existence of a weak spectral gap, a uniform discreteness of the underlying Fuchsian group, and a type of non-accumulation of bounded geodesics, we prove that $\log\detΔ_{2k_n}/\mathrm{vol}(X_{n})$ converges to a constant $C_α$ which depends only on $α=\lim_{n\to\infty} k_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.
