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

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