Theta Functions on Noncommutative Tori

Ordinary theta functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta functions as holomorphic elements of projective modules over noncommutative tori (theta vectors). The theory of these new objects is not only more general, but also much simpler than the theory of ordinary theta-functions. It seems that the theory of theta vectors should be closely related to Manin's theory of quantized theta functions, but we don't analyze this relation.