Abel pairs and modular curves

We consider rational functions on algebraic curves which have a single zero and a single pole. A pair consisting of such a function and a curve is called Abel pair; a special case of an Abel pair is a Belyi pair. In this paper, we study moduli spaces of Abel pairs for curves of genus one. In particular, we compute a number of Belyi pairs over the fields $\mathbb C$ and $\overline{\mathbb F_p}$. This approach could be fruitfully used for the study of Hurwitz spaces and modular curves for fields of finite characteristics.