2026/07/29 by Jay Jorgenson, Lejla Smajlovic, Polyxeni Spilioti
Mathematics · #math.SP #math.NT #msc:11M36 #msc:11F72
arxiv created 2026/07/29 · arxiv updated 2026/07/30
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 Δ2k as well that of the scalar Laplacian Δ0. Our second result studies the asymptotic behavior of the spectral determinant detΔ2kn for a sequence (Xn, χn, kn) for which the genus of Xn 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 logdetΔ2kn/vol(Xn) converges to a constant Cα which depends only on α=limn→∞ kn. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.