On the Action of $\Gamma ^0(N)$ on $\hat{\mathbb{Q}}$

Abstract

In this paper we examine $\Gamma ^0(N)$-orbits on $\hat{\mathbb{Q}}$and the suborbital graphs for $\Gamma ^0(N)$. Each such suborbitalgraph is a disjoint union of subgraphs whose vertices form a blockof imprimitivity for $\Gamma ^0(N)$. Moreover, these subgraphs areshown to be vertex $\Gamma ^0(N)$-transitive and edge $\Gamma^0(N)$-transitive. Finally, necessary and sufficient conditions forbeing self-paired edge are provided.