The graded ring of quantum theta functions for noncommutative torus with real multiplication

Abstract. For the quantum torus generated by unitaries UV = e(θ)V U there exist nontrivial strong Morita autoequivalences in the case when θ is a real qua-dratic irrationality. A.Polishchuk introduced and studied the graded ring of holomorphic sections of powers of the respective bimodule (depending on the choice of a complex structure). We consider a Segre square of this ring whose graded components are spanned by Rieffel scalar products of Polishchuk’s holo-morphic vectors as in [5] and [8]. These graded components are linear spaces of quantum theta functions in the sense of Yu. Manin