Timeline for Outline of the proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite
Current License: CC BY-SA 4.0
8 events
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| Sep 27, 2021 at 12:51 | comment | added | Chris Wuthrich | $E$ is an irreducible curve defined over $K$. Anyway, I recommend to learn basic algebraic geometry: You can't take $R$-valued point of the special fibre, $E(K)/E_0(K)$ is not a scheme, just an abelian group. etc Your "understanding (outline)" above reveals that you confuse a few few things even if you are on the right track to learn it. I think questions like this are more appropriate for math.stackexchange than here. | |
| Sep 27, 2021 at 9:12 | comment | added | Duality | Topology is natural one defined by $v$ on projective space and it's relative topology. Could you explain me more why $E_0(K)$ cannot be considered as $K$-variety ? | |
| Sep 26, 2021 at 11:01 | comment | added | Chris Wuthrich | You really should not post the same question on both at the same time. math.stackexchange.com/questions/4260219/… | |
| Sep 26, 2021 at 10:55 | comment | added | Chris Wuthrich | You say "local field" so I assumed the residue field is finite. You deleted the statement that is is hard to prove, which my first comment was about. Your last paragraph does not make sense. What topology do you put on $E(K)$ and $E_0(K)$? Note there is no $K$-variety $E_0$ whose $K$-rational points are $E_0(K)$. It is just a subgroup of $E(K)$, so clearly the "connected component" argument can't work over $K$, you need $R$ and hence the Néron model. | |
| Sep 26, 2021 at 10:50 | comment | added | Duality | Elementary proof only applies when residue field is finite. In Silverman ( and in this question), residue field is finite is not supposed. And I'm asking whether my understanding of the method using Neron model is correct, not asking what kind of other method are there. | |
| Sep 26, 2021 at 10:09 | history | edited | Duality | CC BY-SA 4.0 |
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| Sep 26, 2021 at 9:11 | comment | added | Chris Wuthrich | Exercise VII.6 in Silverman gives an elementary method to prove that they are finite, but you need to analyse the equation modulo higher powers of the maximal ideal to find the index - and the best way to do that is by Tate's algorithm. The best way to formulate it is using Néron models. The more interesting question is mathoverflow.net/questions/404621 . | |
| Sep 26, 2021 at 6:34 | history | asked | Duality | CC BY-SA 4.0 |