Explicit semi-stable theorem for elliptic curves over $p$-adic fields

Question

In this paper of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v_p(\Delta_E))}$ where $E$ is an elliptic curve over $\mathbb{Q}_p$ and "pgcd" is the greatest common divisor.

Then the authur says on the next paragraph that if $p>5$ and $E$ has potential good reduction over $\mathbb{Q}_p$, then $e=\text{dst}(E)$ is prime to $p$ and $E$ has good reduction on an totally ramified extension of $\mathbb{Q}_p$ of degree $e$.

So the question is how to prove the two propositions and I want to know if there are some reference about the relations between "défaut de semi-stabilité" and the reduction types of elliptic curves.

Thanks!

Answer 1

More generally, let $K/\mathbb{Q}_p$ be a finite extension with $p\ge5$, and let $E/K$ have potential good reduction. Then you can read off the Kodaira-Neron reduction type from the valuation of the minimal discriminant. This is in Tate's table in Antwerp IV, reproduced in my Arithmetic of Elliptic Curves book (Table 15.1) and many other places. Let $\pi$ be a uniformizer and $v$ the normalized valuation on $K$. Take a minimal Weierstrass equation for $E$, $$ E: Y^2 = X^3 + AX + B . $$ To get good reduction, we change variables by $$ E: Y^2 = X^3 + \pi^{-4r}AX + \pi^{-6r}B \quad\text{with}\quad r = \min\left\{\frac14v(A),\frac16v(B)\right\}. $$ So there is a field of good reduction $K_E^{\text{good}}$ given by one of $K(A^{1/4})$ or $K(B^{1/6})$, from which one can read off the ramification index $e(K_E^{\text{good}}/K)$. A case-by-case analysis yields complete information, as follows: $$ \begin{array}{|c|c|c|} \hline \text{Type} & II & III & IV & I_0^* & IV^* &III^* & II^* \\ \hline v(A) & \ge1 & 1 & \ge1 & =2~\text{or}~\ge2 & \ge3 & 3 & \ge4 \\ \hline v(B) & 1 & \ge2 & 2 & \ge3~\text{or}~=3 & 4 & \ge5 & 5 \\ \hline v(\Delta_E) & 2 & 3 & 4 & 6 & 8 & 9 & 10 \\ \hline K_E^{\text{good}} & K(\pi^{1/6}) & K(\pi^{1/4}) & K(\pi^{2/6}) & K(\pi^{2/4})=K(\pi^{3/6}) & K(\pi^{4/6}) & K(\pi^{3/4}) & K(\pi^{5/6}) \\ \hline e(K_E^{\text{good}}/K) & 6 & 4 & 3 & 2 & 3 & 4 & 6 \\ \hline 12/\gcd(12,v(\Delta)) & 6 & 4 & 3 & 2 & 3 & 4 & 6 \\ \hline \end{array} $$