Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume

Abstract

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.