  The paper consists of two sections. The first section provides a new
definition of mirror symmetry of abelian varieties making sense also over
$p$-adic fields. The second section introduces and studies quantized
theta-functions with two-sided multipliers, which are functions on
non-commutative tori. This is an extension of an earlier work by the author. In
the Introduction and in the Appendix the constructions of this paper are put
into a wider context.
