Abel Pairs and Modular Curves

Abstract

Rational functions on algebraic curves, which have a single zero and a single pole, are considered. A pair consisting of such a function and a curve is called an Abel pair; a special case of an Abel pair is a Belyi pair. In the present paper, moduli spaces of Abel pairs for curves of genus one are studied. In particular, a number of Belyi pairs over the fields ℂ and \( \overline{{\mathbb{F}}_p} \) is computed. This approach could be fruitfully used in studying Hurwitz spaces and modular curves for fields of finite characteristics.