= Deep logarithm subgroup of a formal group
{title2=$m>\frac{e}{p-1},\quad F(\pi^m\mathcal O_K)\cong(\mathcal O_K,+)$}
Let $F$ be a <formal group law> over the <valuation ring> of a finite extension $K/\mathbb Q_p$, with $v_K(\pi)=1$ and $e=v_K(p)$. If $m>e/(p-1)$, its <formal logarithm> is an isomorphism $F(\pi^m\mathcal O_K)\to(\pi^m\mathcal O_K,+)$. Indeed, its scaled series is $U+\pi R(U)$ with $R$ integral and convergent on $\mathcal O_K$; solving by contraction produces the <formal group exponential>. The subgroup has index $q^{m-1}$ in $F(\pi\mathcal O_K)$, where $q$ is the residue-field order.
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