Label of a subgroup $\GL_2(\widehat\mathbb Z)$ (awaiting review)

Let $H$ be an open subgroup of $\GL_2(\widehat \Z)$ for which $\det(H)=\widehat \Z^\times$. The label of $H$ has the form $\texttt{N.i.g.n}$, where

• $N$ is the level of $H$,
• $i$ is the index of $H$,
• $g$ is the genus of $H$, and
• $n$ is a positive integer that distinguishes nonconjugate subgroups of the same level, index, and genus.

The positive integer $n$ is defined by ordering the subgroups of $\GL_2(\Z/n\Z)$ as described in Section 2.7 of [arXiv:2106.11141].

When $\det(H)\ne \widehat \Z^\times$ the label of $H$ has the form $\texttt{M.d.m-N.i.g.n}$, where $\texttt{M.d.m}$ identifies the open subgroup $\det(H)\le \GL_1(\widehat \Z)$ in terms of its level $M$, index $d$, and a positive integer $m$ that is defined by ordering the open subgroups of $\GL_1(\widehat \Z)$ via the lexicographic ordering of the Conrey labels in the corresponding group of Dirichlet characters of modulus $M$. In this case the index $i$ is taken relative to the inverse image of $\det(H)$ in $\GL_2(\widehat \Z)$.

This convention is also used to label open subgroups of $\GL_2(\Z_\ell)$ and arbitrary subgroups of $\GL_2(\Z/n\Z)$ by using the label of the inverse image of $H$ in $\GL_2(\widehat \Z)$.