Timeline for Intersection of field extensions of torsion points of non-isogenous elliptic curves
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 9, 2011 at 13:12 | comment | added | David E Speyer |
I think the OP's assumption that the Galois group is $SL_2(\mathbb{Z}_p)$ should force $E$ not to have CM.
|
|
| Sep 9, 2011 at 6:51 | comment | added | Denis Chaperon de Lauzières | Note that Serre had already proved the isogeny theorem for elliptic curves when one of the $j$-invariant is not an integer (in which case the Tate curve gives an "easy" proof...). | |
| Sep 9, 2011 at 6:39 | comment | added | Michael | @ACL, Yes, I understand why you wrote it that way, and it was a useful formulation, my objection was was more a stylistic remark: putting Faltings in parenthesis obscures the actual content. | |
| Sep 9, 2011 at 6:04 | comment | added | ACL | To Michael: I changed hypothesis (ii) of Serre to the one I gave here, just because it is more natural, and Serre explicitly mentioned that point. (He had been able to prove it for non-integral $j$-invariants.) To Eric: You're absolutely right! I overlooked that point. | |
| Sep 8, 2011 at 23:16 | comment | added | Michael | I don't quite understand the first line, because there was no assumption on $K$, so $K$ could equal $K(\zeta_{l^{\infty}})$ (not if $K$ is a number field, of course). It also seems a little strange to phrase the statement of Serre's theorem in the way you do - by far the hardest part of that statement is Faltings contribution (the Tate conjecture). However, you seemed to have mastered the dark art of divining exactly what the OP wanted to know, rather than what they actually asked! | |
| Sep 8, 2011 at 22:18 | vote | accept | Adam Harris | ||
| Sep 8, 2011 at 22:17 | vote | accept | Adam Harris | ||
| Sep 8, 2011 at 22:18 | |||||
| Sep 8, 2011 at 22:17 | comment | added | Adam Harris | Apologies - I forgot to put non-CM curves as a hypothesis in my question (but it's probably not good to change this now?) and $p \geq 5$ was just a guess to keep the question more concise, but the theorem of Serre which ACL quoted is exactly what I wanted so thank you all! | |
| Sep 8, 2011 at 20:52 | comment | added | Erick Knight | and thus $k$ must contain (if someone can edit that in, it would be appreciated) | |
| Sep 8, 2011 at 20:50 | comment | added | Erick Knight | Just a remark: the fact that the Galois group is $\text{SL}_2(\mathbb{Z}_p)$ implies that the $p$-adic cyclotomic character on $G_k$ is trivial, and thus must contain the $p$-adic cyclotoic extension. | |
| Sep 8, 2011 at 20:10 | history | answered | ACL | CC BY-SA 3.0 |