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Modular invariant of quantum tori
Author(s) -
CastañoBernard C.,
Gendron T. M.
Publication year - 2014
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdu016
Subject(s) - invariant (physics) , infimum and supremum , mathematics , modular design , torus , modular invariance , pure mathematics , quadratic equation , quantum , ramanujan's sum , discrete mathematics , mathematical physics , quantum mechanics , computer science , geometry , physics , operating system
The modular invariant j qt of quantum tori is defined as a discontinuous,PGL 2 ( Z ) ‐invariant multi‐valued map of R . For θ ∈ Q it is shown thatj qt ( θ ) = ∞ . For quadratic irrationalities, experiments conducted with the PARI/GP computer algebra system suggest thatj qt ( θ )is a finite set. In the case of the golden mean φ , we produce explicit formulas for the experimental supremum and infimum ofj qt ( φ )involving weighted versions of the Rogers–Ramanujan functions. Finally, we define a universal modular invariant as a continuous and single‐valued map of ultrasolenoids from which j qt as well as the classical modular invariant of elliptic curves may be recovered as subquotients.