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In Silverman's Arithmetic of Elliptic Curves, in Chapter VII, Exercise 7.5, one is asked to show that three elliptic curves have good reduction over given p-adic field extensions (where they have bad reduction over the base p-adic field). Specifically one is asked to show good reduction by "writing down"(!) a minmal Weirstrauss equation over the field.

However, both footling around doing Tate's Algorithm by hand, and kicking it into Magma seem to suggest that the first two don't actually have good reduction in the given fields, which seems curious.

The curves and extensions are as follows:

  1. $E: y^2 = x^3 + x,\text{ defined over }\mathbb{Q}_2(\pi),\pi^8=2$
  2. $E: y^2 +y = x^3,\text{ defined over }\mathbb{Q}_3(\pi),\pi^4=3$
  3. $E: y^2 = x^3 + x^2 -3x-2 ,\text{ defined over }\mathbb{Q}_5(\pi),\pi^4=5$

Here is the Magma code I am executing for the first one:

K:=pAdicField(2,400);
R<x>:=PolynomialRing(K);
E:=EllipticCurve([K|0,0,0,1,0]);
new:=x^8-2;
F:=SplittingField(new);
EF:=BaseChange(E,F);
loc,min:=LocalInformation(EF);
loc;

And its output: <F.1 + O(F.1^6401), 12, 6, 4, I2*, true>.

So this would seem to suggest there is some measure of error in the text: but I thought it was more likely that I was missing something. I would be interested to hear what people think.

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  • $\begingroup$ To be precise, you have copied the formulation of question 7.5 as in the first edition. In the second edition the question is over the splitting fields of the polynomials $x^8-2$ and $x^4-3$ for 1) and 2). $\endgroup$ – Chris Wuthrich Oct 5 '16 at 22:04
  • $\begingroup$ Using sage, I also get for 1) over $\mathbb{Q}_2(\pi)$ type III* and type I2* over the Galois closure. For 2) type IV* over $\mathbb{Q}_3(\pi)$ and its Galois closure. 3) instead is really good over $\mathbb{Q}_5(\pi)$. Though, I have not thought about whether I should believe any of this. $\endgroup$ – Chris Wuthrich Oct 5 '16 at 22:08
  • $\begingroup$ Chris: interesting; I have the hardback first edition, which is as you describe, and then I copied this from a digital version which I have, which I think is based off a softback version of the same edition. I don't believe I have access to the second edition, but it sounds like it as it is in the first. $\endgroup$ – Wren Robson Oct 6 '16 at 12:18
  • $\begingroup$ My quick computation, that may be wrong, would tell me that for 1), the Serre-Tate group is $S_3$ and for 2) it is $D_4$. So just making $\Delta$ a 12-th power can indeed not help in this case. I think it would be good to send the author an email in case he missed this post here. It doesn't figure in the erratum to his book. $\endgroup$ – Chris Wuthrich Oct 7 '16 at 10:09
  • $\begingroup$ Good idea, I'll do just that. $\endgroup$ – Wren Robson Oct 10 '16 at 22:26
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In the 2nd edition of The Arithmetic of Elliptic Curves, the problem is stated as follows:

Exercise 7.5 Show that the following elliptic curves have good reduction over a field of the indicated form by writing down a minimal equation for $E$ over that field.

(a) $E:y^2=x^3+x$, $\mathbb{Q}_2(\eta,i)$, $\eta^8=2$, $i^2=-1$.

(b) $E:y^2+y=x^3$, $\mathbb{Q}_3(\pi,\eta)$, $\pi^2=\sqrt{-3}$, $\eta^3=2$.

(c) $E:y^2=x^3+x^2-3x-2$, $\mathbb{Q}_5(\pi)$, $\pi^4=5$.

It is easy enough to do (b) by hand. We substitute $x\to \pi^2 X+\eta$ and $y\to \pi^3 Y+1$ and divide by $\pi^6$ to obtain the Weierstrass equation $$ Y^2 - \pi Y = X^3 - \eta\pi^2 X^2 + \eta^2X. $$ This Weierstrass equation has discriminant $\Delta=1$, so in fact represents a global everywhere-good-reduction model for $E$ over the field $\mathbb{Q}(\sqrt[4]{-3},\sqrt[3]{2})$.

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