Modular Invariant of Quantum Tori

The quantum modular invariant jqt(θ) of θ ∈R is defined as a discontinuous PGL2(Z)-invariant multi-valued map using the distance-to-the-nearest-integer function ‖ · ‖. For θ ∈Q it is shown that jqt(θ) =∞ and for quadratic irrationalities PARI/GP experiments suggest that jqt(θ) is a finite set. In the case of the golden mean φ, we produce explicit formulas involving weighted versions of the RogersRamanujan functions for the experimental supremum and infimum. We then define a universal modular invariant ¦ j : ¦  Mod→ ¦Ĉ as a continuous and single valued map of ultrasolenoids, such that 1) the classical modular invariant is a quotient of the restriction of ¦ j to a subsolenoid Modcl ⊂ ¦  Mod fibering over the classical moduli space of elliptic curves and 2) the quantum modular invariant is a quotient of the restriction of ¦ j to a subsolenoid Modqt ⊂ ¦  Mod fibering over the moduli space of elliptic curves equipped with a Kronecker foliation.