Resolvent Trace Formula and Determinants of 𝑛 Laplacians on Orbifold Riemann Surfaces

Abstract

For 𝑛 a nonnegative integer, we consider the 𝑛-Laplacian Δₙ 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 Δₙ + 𝑠(𝑠 + 2𝑛 − 1), from which we deduce the regularized determinant of Δₙ, denoted by det′Δₙ. Taking into account the contribution from the absolutely continuous spectrum, det′Δₙ is equal to a constant Cₙ times 𝛧(𝑛) when 𝑛 ≥ 2. Here 𝛧(𝑠) is the Selberg zeta function of 𝑋. When 𝑛 = 0 or 𝑛 = 1, 𝛧(𝑛) is replaced by the leading coefficient of the Taylor expansion of 𝛧(𝑠) around 𝑠 = 0 and 𝑠 = 1, respectively. The constants Cn are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but are independent of the moduli parameters.