Timeline for Elliptic curves with potential good reduction over a prescribed extension
Current License: CC BY-SA 3.0
6 events
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| Nov 14, 2015 at 16:43 | comment | added | Jackson Morrow | @eric Thank you for you comments. After running through some examples, I do agree with you. In above mentioned work, Frey only considers the elliptic curve $y^2 = x^3 + 1$ for $p=3$, which is why I wrote that it is dealt with separately. However, it seems as though this is a special case. | |
| Nov 14, 2015 at 0:15 | comment | added | eric | If you want to find an explicit example just browse through some table of elliptic curves trying the ones with integral j-invariant at 3 and conductor a multiple of 3, and then just compute how 3 ramifies in the field cut out by the 2-torsion. The moment you find an example where inertia has order a multiple of 3 you might be in trouble. | |
| Nov 13, 2015 at 21:20 | comment | added | eric | I shouldn't think one can do this for p=3 in general. If the image of inertia at 3 in the ell-adic representation has order a multiple of 3 (and I don't see why this can't happen) then you'll need an extension of order a multiple of 3 to kill it. | |
| Nov 13, 2015 at 21:10 | comment | added | Jackson Morrow | @VesselinDimitrov Thank you for the comment. This does help me for the cases when $p = 5$ or $7$, however, I am still a bit confused about the case when $p=3$. Under my assumption, the elliptic curve $E/K$ has a $K$-rational points of order 3. Since $K(E[3]) \subseteq K(E[12])$, this would imply that $3$ divides the degree of the normal extension over which $E$ acquires stable reduction. | |
| Nov 13, 2015 at 19:53 | comment | added | Vesselin Dimitrov | Every abelian variety $A/K$ acquires stable reduction over the normal extension $K(A[12])/K$. This is an extension of bounded degree, when $\dim{A}$ is fixed. | |
| Nov 13, 2015 at 16:23 | history | asked | Jackson Morrow | CC BY-SA 3.0 |