Agreeable (awaiting review)

An open subgroup $H\le \GL_2(\widehat\Z)$ (and the corresponding modular curve $X_H$) is agreeable (as defined by Zywina in Section 1.4 of [arXiv:2206.14959]) if the following hold:

• $\det(H)=\widehat\Z^\times$;
• $H$ contains the subgroup of $\GL_2(\widehat \Z)$ formed by scalar matrices;
• the level of $H\le \GL_2(\widehat\Z)$ and the level of $H\cap \SL_2(\widehat \Z)\le \SL_2(\widehat \Z)$ have the same odd prime divisors.

The agreeable closure $\widetilde H$ of $H$ is the intersection of all the agreeable open subgroups of $\GL_2(\widehat \Z)$ that contain $H$.

The agreeable closure $\widetilde H$ contains $H$ as a normal subgroup and has the same commutator subgroup as $H$; see Proposition 8.1 of [arXiv:2206.14959]. It follows that the quotient $\widetilde{H}/H$ (called the agreeable quotient) is finite abelian.