In 1977 at the Cortona Conference on several complex variables the speaker proposed to make studies on "the K3 modular function". That is the explicit expression of the inverse of the period map for a family of K3 surfaces. In this talk, we give a short summary of our study on this direction for 45 years. A recent result is based on the studies by Prof. Salvati-Manni and others. You can see the outline of the talk by two papers by A. Nagano and H. Shiga on Transformation Groups (2022) and Math. Nachrichten (2022).