Jerome Dubois and Stavros Garoufalidis
Abstract: If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation $\rho_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of $\rho_0$ in $X_M$ and observed that it is an analytic map. In the present paper we prove that this non-abelian Reidemeister torsion is the germ of a unique rational function on $X_M$. This implies the existence of a polynomial relation of the torsion of a representation and the trace of the meridian, and explains why the torsion of the discrete faithful representation lies in the trace field of $M$. We postulate that the coefficients of the $1/N^k$-asymptotics of the Parametrized Volume Conjecture for $M$ are elements of the field of rational functions on $X_M$.
Key words: knots, $A$-polynomial, $\tau$-polynomial, Reidemeister torsion, volume, character variety, 3-manifolds, hyperbolic geometry, invariant trace field.
Notes: 11 pages, 0 figures.