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

For the quantum torus generated by unitaries UV = e(θ)VU there exist nontrivial strong Morita autoequivalences in the case when θ is a real quadratic 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 holomorphic vectors. These graded components are linear spaces of quantum theta functions in the sense of Yu. Manin.