# Blog Archives

# Topic Archive: ultrapowers

# Modular Invariant of Quantum Tori

The modular invariant *j ^{qt}* of quantum tori is defined as a discontinuous,

*-invariant multi-valued map of the reals*

**PGL(2,Z)****. For**

*R**θ ∈*

**Q**,

*j*and for quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest that

^{qt}(θ) = ∞*j*is a finite set. In the case of the golden mean φ, we produce explicit formulas for the experimental supremum and infimum of

^{qt}(θ)*j*involving weighted generalizations of the Rogersâ€“Ramanujan functions. Finally, we define a universal modular invariant as a continuous and single-valued map of “ultrasolenoids” (quotients of sheaves of ultrapowers over Stone spaces) from which

^{qt}(φ)*j*as well as the classical modular invariant of elliptic curves may be recovered as subquotients.

^{qt}