Genus $0$ Modular curves of prime power level with a point defined over number fields other than $\Q$

TL;DR

This paper classifies all genus 0 modular curves of prime power level with a rational point, associated to certain subgroups of the Galois group, over number fields other than .

Contribution

It provides a complete classification of prime power level genus 0 modular curves with a point over number fields other than .

Findings

List of all such modular curves is provided.

Characterization of the associated subgroups of .

Conditions under which these curves have points over specific number fields.

Abstract

Associated to an open subgroup $G$ of $\GL_2(\Zhat)$ satisfying conditions $-I \in G$ and $\det(G) \subsetneq (\Zhat)^{\times}$ there is a modular curve $X_G$ which is a smooth compact curve defined over an extension of $\Q.$ In this article, we give a complete list of all such prime power level genus $0$ modular curves with a point.