Pro-\(p\) criterion of good reduction of punctured elliptic curves (Q6143269)

Summary: In this paper we consider an affine hyperbolic curve \(X\) over a \(p\)-adic local field \(K\) obtained from an elliptic curve \(E\) by removing a \(K\)-rational point given by the origin \(O\). We introduce the maximal geometrically pro-\(p\) quotient \(\Pi_X\) of the étale fundamental group \(\pi_1^{\text{ét}} (X)\) of the hyperbolic curve \(X\) and analyse the problem of determining the reduction type of the elliptic curve \(E\) from the topological group \(\Pi_X\). As our main result we give a group-theoretic construction of the reduction type of \(E\) from the group \(\Pi_X\) under the assumption that \(p \geq 5\) and that \(E\) has a nontrivial \(K\)-rational \(p\)-torsion point.