= Torsion test using a deep formal logarithm subgroup
{title2=$\log_F([N]P)\in Np^m\mathbb Z_p$}
Let $E/\mathbb Q_p$ have <good reduction>, and choose an integer $m>1/(p-1)$. Its <filtration of elliptic-curve points over a local field> has a <torsion-free group> $E_m$ identified by the <formal logarithm> with $p^m\mathbb Z_p$. The <quotient group> $G=E(\mathbb Q_p)/E_m$ is finite. For $g\in G$ of <order of a group element> $N$, choose a lift $P$. There is a <torsion point of an elliptic curve> in this coset exactly when the displayed condition holds. Indeed, it is equivalent to the existence of $Q\in E_m$ with $[N]Q=[N]P$. Then $P-Q$ is killed by $N$; conversely every torsion representative has order $N$, since $E_m$ is torsion-free. The logarithmic division and the <formal group exponential> give the unique such $Q$. Changing the lift changes the logarithm by an element of $Np^m\mathbb Z_p$, so the criterion is well-defined. This finite-coset test includes residue-characteristic torsion; only the <prime-to-p torsion lifts uniquely at good reduction> are automatic.
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