Seminar in Algebraic Geometry and Number Theory

Abstract: We study the Ruelle zeta function associated to cofinite hyperbolic Riemann surfaces with ramification points twisted by a unitary multiplier system of arbitrary real weight 2k. Using the properties of the zeta-regularized determinants we derive a new form of a functional equation for this function and compute explicitly the first term in the Laurent or Taylor series expansion of this function at s=0. When k is an integer, the order of a zero or a pole at s=0 is expressed in terms of topological invariants of the surface, degrees of singularity of the multiplier system at cusps and torsion points and the order of a zero/pole at s=0 of the scattering determinant. When k is non-integral, the order of a zero or a pole depends only on the  order of a zero/pole at s=0 of the scattering determinant.

We develop three applications that will be discussed during the talk. Two are related to the properties of the higher-dimensional Rademeister torsion for cocompact groups and one is related to evaluation of the lead term at s=0  in terms of special values of L-functions, when the underlying Fuchsian group is arithmetic and not cocompact. For example,  we show that the inverse of the higher-dimensional Rademeister torsion for a general Seifert fibered space and any acyclic irreducible representation of SL(n,C) equals the absolute value of the Ruelle zeta function at s = 0 on the cocompact Fuchsian group associated to this Seifert space and twisted by the n dimensional, weight 2k = 1 unitary multiplier system with the action on ramification points induced by this representation.

This work is joint with Jay Jorgenson and Min Lee.