Abel pairs and modular curves

Abstract

In this article we consider rational functions on algebraic curves, which have one zero and one pole (and call pair of such function and curve Abel pair). We investigate moduli spaces of such functions on curves of genus one; the number of Belyi pairs among them is calculated for fields $\mathbb C$ and $\overline{\mathbb F_p}$. This result could be fruitfully used for investigation of Hurwitz's space and modular curves for fields of finite characteristic.