Real quadratic fields and finite quantum dilogarithms II: traces of quantum modular cocycles

We study traces of quantum modular cocycles at fixed points of hyperbolic elements γ₂(Z), formulate an algebraicity conjecture for these traces, and prove it for the cocycles associated to 1-dimensional state-integrals. In addition, we analyse a finite TQFT associated with $γ$ and conjecture that its invariants for 3-manifolds agree with the traces of their sl₂ quantum modular cocycles. The finite TQFTs predict a yet-to-be-defined motive with cohomology equipped with a Z/2Z-degree grading and a Z-weight grading coming from a version of the Riemann hypothesis, which suggests a similar grading should exist for an extension of the Kauffman bracket skein module, for the framed SL₂(C) Floer cohomology of Abouzaid--Manolescu, and for the quantum modular cocycle. This leads to surprising additive identities among algebraic units, which we confirm for the complement of the simplest hyperbolic knot 4₁, as well as for S¹ × Σg for a closed genus g surface.