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I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension such that the torsion subgroups $E[m]$ and $E[n]$ are contained in $E(L)$. Then $E_L$ has everywhere semistable reduction.

This follows easily from Raynaud's criterion [SGA 7 I, Exp. IX, Proposition 4.7], but I cannot pin down the first appearance of the above version of semistability for elliptic curves.

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  • $\begingroup$ Ask Ogg, Tate, and/or Raynaud. The idea probably doesn't predate SGA7, though once one has the idea there is an "elementary" proof for elliptic curves. You're trying to rule out additive reduction over a local field $F$ when $E[\ell^e]$ is $F$-split for a prime $\ell$ invertible in the residue field with $\ell^e > 2$. Quadratic twist of a Tate curve is ruled out since $\ell^e > 2$. For potential good reduction inertial action is finite, so just need kernel of ${\rm{GL}}_2(\mathbf{Z}_{\ell}) \rightarrow {\rm{GL}}_2(\mathbf{Z}/(\ell^e))$ to be torsion-free, which is clear since $\ell^e>2$. $\endgroup$
    – nfdc23
    Apr 17, 2016 at 6:02

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