Timeline for The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2022 at 9:01 | comment | added | Chris Wuthrich | $E(L)$ is a fin. gen. free $\mathbb{Z}$-module, so as a $\mathbb{Z}[G]$-module it will decompose as a direct sum of $\mathbb{Z}$ with trivial action, $\mathbb{Z}$ with action by $-1$, and $\mathbb{Z}[G]$. It is easy to calculate their cohomology. The local term are only interesting when $v$ isn't split. Now $E(L_v)$ is a fin. gen. $\mathbb{Z}_{\ell}$-module where $\ell$ is below $v$. If $\ell\neq 2$ or if it is torsion-free, the cohomology can be determined easily, otherwise it might be some extra work. | |
| Apr 16, 2022 at 8:30 | comment | added | A. Maarefparvar | @ChrisWuthrich Many thanks for the useful comments. Considering a quadratic extension $L/K$ (in the simplest case), and assuming $E(L)_{tors}=0$, can we determine explicitly the kernel and cokernel of $\mathcal{F}$? Based on your comment: "For small G, one can often calculate all the terms involved in this map explicitly ...", would you please hint me how can I do that for the quadratic extensions? | |
| Apr 16, 2022 at 8:28 | comment | added | A. Maarefparvar | @Chris Wuthrich | |
| Apr 12, 2022 at 17:18 | comment | added | Chris Wuthrich | 1) For small $G$, one can often calculate all the terms involved in this map explicitly even when the rank is positive. I doubt that the target is $\bigoplus$, I think it is $\prod$. The kernel, like the source is a finite group, often rather small, the target and the cokernel can be quite large. 4) I am quite sure it is not zero usually. But I am on holidays so I won't calculate some examples. Maybe later. | |
| Apr 12, 2022 at 15:01 | comment | added | Chris Wuthrich | 2.) The shape of the Galois group may help to determine if the torsion part of $E(L)$ is trivial, but there is little hope to finding a general criteria to impose that the rank does not grow. 3.) No if there is no connection between $L$ and $E$, they could be anything. | |
| Apr 12, 2022 at 14:15 | history | asked | A. Maarefparvar | CC BY-SA 4.0 |