Rationality of the Sl(2,c)-reidemeister Torsion in Dimension 3

If M is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component XM of its SL(2,C)-character variety is an affine complex curve, which is smooth at the discrete faithful representation ρ0. Porti defined a non-abelian Reidemeister torsion in a neighborhood of ρ0 in XM and observed that it is an analytic map, which is the germ of a unique rational function on XM . In the present paper we prove that (a) the torsion of a representation lies in at most quadratic extension of the invariant trace field of the representation, and (b) the existence of a polynomial relation of the torsion of a representation and the trace of the meridian or the longitude. We postulate that the coefficients of the 1/N-asymptotics of the Parametrized Volume Conjecture for M are elements of the field of rational functions on XM .