Modular Invariant of Quantum Tori
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico
The modular invariant jqt of quantum tori is defined as a discontinuous, PGL(2,Z)-invariant multi-valued map of the reals R. For θ ∈ Q, jqt(θ) = ∞ and for quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest that jqt(θ) is a finite set. In the case of the golden mean φ, we produce explicit formulas for the experimental supremum and infimum of jqt(φ) involving weighted generalizations of the Rogers–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 jqt as well as the classical modular invariant of elliptic curves may be recovered as subquotients.