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Let $N \ge 5$ be a prime number and $E/ \mathbb{Q}_N$ be an elliptic curve with additive reduction. Then it is easy to see that there exists a finite extension $K$ over which $E$ has stable reduction.

I want to show that we can choose $K$ so that the ramification index of $K/ \mathbb{Q}_N \le 6$.

The proof of VII 5.5 of Silverman’s AEC does not include that.

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  • $\begingroup$ I'm no expert on elliptic curves, so maybe I am completely mistaken, but doesn't Theorem 5.1.18(2) in Fresnel, van der Put "Rigid analytic geometry and its applications" imply that we can find a $K$ with $[K:\mathbf{Q}_N]\leq 2$? Namely, over a degree $\leq 2$ extension the curve admits a Tate uniformization, and Tate curves are semistable? $\endgroup$ Jan 30 at 16:24
  • $\begingroup$ @Piotr That would be the case of multiplicative reduction not additive. $\endgroup$ Jan 30 at 18:32
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This minimal ramification index is the order of the Serre-Tate group $\Phi$, defined in their article "Good reduction of abelian varieties". It is shown in the proof of theorem 2 there that $\Phi$ is a subgroup of the automorphism group of the reduced elliptic curve over the larger field. If the residual characteristic is not 2 or 3 then the automorphism group is cyclic of order 2, 4 or 6. This gives you what you wanted. In fact $\Phi$ is cyclic of order 2 if the reduction type is I*${}_n$, it is cyclic of order 3 for type IV and IV*, cyclic of order 4 for type III and III* and cyclic of order 6 for type II and II*. For residual characteristic 2 or 3 it is all far more complicated.

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  • $\begingroup$ Thank you very much. But what is the Serre Tate grouop? I cant’t find it. And the theorem treats only potentially good case. $\endgroup$
    – k.j.
    Jan 30 at 19:19
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    $\begingroup$ $\Phi$ is denoted by $G$ in the proof I referenced. It is one of the main subjects of that article, so that would be a good place to start learning about it. The case of potentially multiplicative reduction is easier to explain; see for instance Section 5.6 in "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques" by Serre. I used the notation $\Phi$ from there. $\endgroup$ Jan 30 at 19:30
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    $\begingroup$ The general case is explained in: A. Kraus Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive Manuscripta Math., 69 (1990), pp. 353-385 eudml.org/doc/155566 $\endgroup$
    – Xarles
    Jan 30 at 20:01

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