let $H$ be a closed subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$. Denote by $H_\ell$ the reduction of $H$ modulo $\ell$ and suppose that $H_\ell$ contains a scalar matrix $\overline{\lambda} \operatorname{Id}$. Then $H$ contains a scalar matrix $\lambda \operatorname{Id}$ for some $\lambda \in {\mathbb{Z}}_\ell^\times$ with $\lambda \equiv \overline{\lambda} \pmod{\ell}$.

Let $h \in H$ be any element that is congruent modulo $\ell$ to $\overline{\lambda} \operatorname{Id}$. Let $\lambda \in {\mathbb{Z}}_\ell^\times$ be the Teichmüller lift of $\overline{\lambda}$ (that is, $\lambda^\ell=\ell$ and $\lambda \equiv \overline{\lambda} \pmod{\ell}$) and write $h=\lambda h_1$, where $h_1 =\operatorname{Id}+\ell A$ for some $A\in\operatorname{Mat}_2({\mathbb{Z}}_\ell)$. The sequence $h^{\ell^n} = \lambda^{\ell^n} h_1^{\ell^n}= \lambda h_1^{\ell^n}$ converges to $\lambda \operatorname{Id}$, because for every $n$ we have $h_1^{\ell^n} = \left( \operatorname{Id} + \ell A \right)^{\ell^n} \equiv \operatorname{Id} \pmod{\ell^n}$. As $H$ is closed, the limit of this sequence, namely $\lambda \operatorname{Id}$, also belongs to $H$ as claimed.

The $\ell$-adic failure {#sec:lAdicFailure}
=======================

The aim of this section is to study the $\ell$-adic failure $A_\ell(N)$ for a fixed prime $\ell$. The divisibility properties of $\alpha$ in the group $E(K)$ play a crucial role in the study of this quantity, so we begin with the following definition:

\[def-divisib\] Let $\alpha \in E(K)$ and let $n$ be a positive integer. We say that $\alpha$ 