Abel pairs and modular curves

TL;DR

This paper studies Abel pairs, which are rational functions with one zero and one pole on algebraic curves, focusing on genus one curves and their moduli spaces, with applications to Belyi pairs and modular curves over various fields.

Contribution

It investigates the moduli spaces of Abel pairs on genus one curves and calculates the number of Belyi pairs over complex and finite fields, linking to Hurwitz and modular curves.

Findings

Number of Belyi pairs on genus one curves over a9 and a4p fields calculated

Moduli spaces of Abel pairs characterized for genus one curves

Results applicable to Hurwitz's space and modular curves in finite characteristic

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.