Winding quotients for virtual period maps of rank 1

We illustrate a rank 1 model of virtual period maps and their associated winding quotient, where the winding quotient is a new phenomenon appeared in a recent study of virtual period maps and it requires a reformulation of the classical Jacobi inversion problem for the period maps due to the appearance of exponents which are imaginary numbers. We answer to the new inversion problem by introducing the q-multiplicatively periodic function, whose pull-back to the winding covering space is the Weierstrass p-function up to a correction by Eisenstein series E2. The function appears also in the study of mathematical physics as the propagator on elliptic curves.