Timeline for Interpretation of $H^3(\mathrm{Gal}(L/K),L^\times)$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 16 at 17:44 | comment | added | Firebolt2222 | I think I found a nice paper (jstor.org/stable/2374333?seq=2) and a suitabale interpretation, which I will have to check myself a bit more carefully. But for other people that are interested, I will write a more elaborate answer as soon as I get a better grasp of it. | |
| May 16 at 17:42 | comment | added | Firebolt2222 | @DanielLoughran Thank you very much for the reference! This seems very helpful. I'll have a deeper look into it. Though I have to admit that I'm not that strong in algebraic geometry. | |
| S May 15 at 14:07 | history | bounty ended | CommunityBot | ||
| S May 15 at 14:07 | history | notice removed | CommunityBot | ||
| May 7 at 19:26 | comment | added | Daniel Loughran | Elements of this form appear in the paper "Surfaces defined by pairs of polynomials" by Skorobogatov and Gvirtz-Chen. it seems in general to get something non-trivial you need to be using transcendental elements (indeed this cohomology group is trivial over number fields). | |
| May 7 at 19:25 | comment | added | Daniel Loughran | At least if you have the corresponding roots of unity in our ground field, then you can construct elements of this Galois cohomology group using the cup product $H^1(k,\mu_n) \times H^1(k,\mu_n) \times H^1(k,\mu_n) \to H^3(k,\mu_n)$, which can be related to your cohomology using Kummer theory. Whether such an element represents a non-trivial cohomology class is a tricky question. | |
| S May 7 at 12:57 | history | bounty started | Firebolt2222 | ||
| S May 7 at 12:57 | history | notice added | Firebolt2222 | Draw attention | |
| May 2 at 12:17 | comment | added | Firebolt2222 | @Bma Thank you I'll have a look | |
| May 1 at 7:43 | comment | added | Bma | Arithmetic Duality Theorems by Milne may have some relevant material | |
| Apr 30 at 14:48 | history | edited | Firebolt2222 | CC BY-SA 4.0 |
added 30 characters in body
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| S Apr 30 at 14:42 | review | First questions | |||
| Apr 30 at 14:55 | |||||
| S Apr 30 at 14:42 | history | asked | Firebolt2222 | CC BY-SA 4.0 |