2
$\begingroup$

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!

$\endgroup$
7
$\begingroup$

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: $$ \begin{array}{|c|c|c|} \hline \text{Type} & II & III & IV & I_0^* & IV^* &III^* & II^* \\ \hline v(A) & \ge1 & 1 & \ge1 & =2~\text{or}~\ge2 & \ge3 & 3 & \ge4 \\ \hline v(B) & 1 & \ge2 & 2 & \ge3~\text{or}~=3 & 4 & \ge5 & 5 \\ \hline v(\Delta_E) & 2 & 3 & 4 & 6 & 8 & 9 & 10 \\ \hline K_E^{\text{good}} & K(\pi^{1/6}) & K(\pi^{1/4}) & K(\pi^{2/6}) & K(\pi^{2/4})=K(\pi^{3/6}) & K(\pi^{4/6}) & K(\pi^{3/4}) & K(\pi^{5/6}) \\ \hline e(K_E^{\text{good}}/K) & 6 & 4 & 3 & 2 & 3 & 4 & 6 \\ \hline 12/\gcd(12,v(\Delta)) & 6 & 4 & 3 & 2 & 3 & 4 & 6 \\ \hline \end{array} $$

$\endgroup$
  • $\begingroup$ Thanks for your answers. I see the table 4.1 on page 365 of your book "advanced topics...". But I’m still a little confused. When we are given an elliptic curve over a p-adic field, and if we have computed its Kodaira-Neron reduction type, so we know its discriminant, its conductor and its behavior of $j$-invariant, but how to know the ramification needed to acquire good reduction? Where the "dst" come from? Sorry to disturb you again. $\endgroup$ – Sssss Oct 14 at 15:54
  • 1
    $\begingroup$ @Sssss Since $p\ge5$, WLOG we have $E:y^2=X^3+AX+B$. Each reduction type corresponds to some valuation information on $A$ and $B$. I'll do one example. Type III is $v(A)=1$ and $v(B)\ge2$ to get $v(\Delta)=3$. To get good reduction, we go to $Y^2=X^3+\pi^{-4}AX+\pi^{-6}B$, and we need $v(\pi^{-4}A)=0$, so get good reduction over $\mathbb Q_p(A^{1/4})$. Since $v(A)=1$, the ramification index is $4$. The other cases can be handled similarly. I don't know if there's a simpler way to do all cases simultaneously. $\endgroup$ – Joe Silverman Oct 14 at 17:11
  • $\begingroup$ Thanks! I always enjoy reading both your books and your answers on MO. $\endgroup$ – Sssss Oct 15 at 12:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.