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

Abstract

Let $(X,\chi ,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $\chi$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,\chi ,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $\chi$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian ${\Delta }_{2k}$ as well that of the scalar Laplacian ${\Delta }_0$. Our second result studies the asymptotic behavior of the spectral determinant $\det {\Delta }_{2k_n}$ for a sequence $(X_n,{\chi }_n,k_n)$ 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 $\log \det {\Delta }_{2k_n}/\mathrm{vol}(X_n)$ converges to a constant $C_{\alpha }$ which depends only on $\alpha ={\lim }_{n\to \infty }{k}_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.