Ramification index and additive reduction of elliptic curves

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.

• 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? Jan 30 '21 at 16:24
• @Piotr That would be the case of multiplicative reduction not additive. Jan 30 '21 at 18:32

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.
• $\Phi$ 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. Jan 30 '21 at 19:30