Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

This work is a first step towards a theory of $q$-deformed complex numbers. Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials.