= Unramified division torsors at good primes
{title2=$v\nmid m,\quad E\text{ good at }v$}
= Unramified division points at good reduction
{synonym}
At a prime of <good reduction of an elliptic curve> with residue characteristic not dividing $m$, multiplication by $m$ extends to a finite étale map of the smooth proper elliptic group scheme over the valuation ring. A local rational point extends to a section by properness. Its division fibre is therefore a finite étale torsor, so the <Kummer map of an elliptic curve> gives an unramified cohomology class. This bounds ramification in the <Weak Mordell-Weil theorem>.
A direct proof describes the coordinates, not only the cohomology class. Reduce $P$ by properness and choose a point $\overline Q$ with $[m]\overline Q=\widetilde P$ over a finite extension of the <residue field>. Over the corresponding <unramified extension> of the complete <local field>, lift $\overline Q$ by the <Hensel lemma>. The error $[m]Q_0-P$ is in the <kernel of reduction of an elliptic curve>; <prime-to-residue-characteristic multiplication on a formal group> corrects it uniquely. There is no $m$-torsion in that kernel, even after a ramified field extension, so every division point with that reduction equals this unramified lift. Consequently all <division points of an elliptic curve> above $P$ have unramified coordinate fields. The torsion case is discussed in https://www.jmilne.org/math/Books/EC2.pdf[J. S. Milne's Elliptic Curves, Chapter V, Proposition 8.1].
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