This paper attempts to compute the adjoint Reidemeister torsion for a cusped hyperbolic \(3\)-manifold from its ideal triangulation. We need a cell decomposition of a 3-manifold \(M\) and a homomorphism from \(\pi_1(M)\) into a linear group to define the Reidemeister torsion. Furthermore, the adjoint Reidemeister torsion requires cohomology classes in the local system given by a homomorphism of \(\pi_1(M)\). This paper considers the cell decomposition induced from an ideal triangulation and the holonomy representation \(\rho\) of a hyperbolic \(3\)-manifold \(M\), which gives the identification of \(M\) with \(\rho(\pi_1(M)) \backslash \mathbb{H}^3\). Here the holonomy representation \(\rho\) is an embedding into \(\mathrm{PSL}_2(\mathbb{C}) \simeq \mathrm{Isom}^+\,\mathbb{H}^3\).
The author shows that the adjoint Reidemeister torsion of a cusped hyperbolic \(3\)-manifold \(M\) turns into the product of two Reidemeister torsions. One is the Reidemeister torsion of the long exact sequence of the local systems for the pair of a finite cell complex \(X\) and its 1-skeleton \(X^{(1)}\). The other is that of the \(1\)-skeleton \(X^{(1)}\). Here \(X\) denotes a finite cell complex given by dual cells for an ideal triangulation of \(M\). The main results reveal that the Reidemeister torsion of the long exact sequence of the local systems reduces to that of the “tangential exact sequence of infinitesimal gluing equations” which are the derivatives of the hyperbolic and completeness equations for an ideal triangulation. The author made this reduction according to the identification between the first cohomology group in the local system and the tangent space over the character variety of \(\pi_1(M)\). The author also shows that the Reidemeister torsion for the “tangential exact sequence of infinitesimal gluing equations” is given by the enhanced Neumann-Zagier datum for the hyperbolic structure of \(M\). This paper conjectures that the Reidemeister torsion of the 1-skeleton \(X^{(1)}\) is also a quantity given by the enhanced Neumann-Zagier datum, which leads us to an affirmative answer for the generalized 1-loop conjecture. The original 1-loop conjecture states that the Reidemeister torsion for a hyperbolic \(3\)-manifold with one cusp is the 1-loop part of a formal power series defined by the enhanced Neumann-Zagier datum, which is due to T. Dimofte and S. Garoufalidis [Geom. Topol. 17, No. 3, 1253–1315 (2013; Zbl 1283.57017)]. We can find the supporting example of the generalized 1-loop conjecture in the Appendix. As the supporting example, the author computes the Reidemeister torsion of the 1-skeleton \(X^{(1)}\) for the sister manifold of the figure eight knot complement.