Timeline for Preimage of projection of idèles, and other usual maps
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2016 at 21:54 | comment | added | znt | No your exact sequence above is not correct because there are issues with global units; the first map is not injective (and it also seems to assume your number field is imaginary quadratic, something which isn't mentioned in the question as it stands). Honestly, nobody wants to answer questions which are written in comments. Extended chat is discouraged in comments. Please edit your question; find out something precise which you don't understand and then try and ask that. Or ask a new question. | |
| Dec 13, 2016 at 12:58 | comment | added | Desiderius Severus | @znt If the exact sequence above is correct, am I right if I consider the subset $\{(z,x) \in \mathbf{C}^\times \times \hat{\mathcal{O}}^\times \ : \ x \mod \mathfrak{m} \in Im(Z/\mathfrak{m}\cap Z \to O_K/\mathfrak{m})\} \subset \mathbf{C}^\times \times \hat{\mathcal{O}}^\times \subset A_K^\times/K^\times$ ? I would like to determine if my idèle built from $\mathfrak{a}$ is in it (that is, such an $x$), or not. Does it make sense? | |
| Dec 13, 2016 at 12:48 | comment | added | znt | It is difficult to understand your question because, at the time of writing my comment, your question still has errors in. There is an issue with global units and an issue with the class group, and you might be lost but I cannot help because I still do not understand the question. Can you fix it? I still don't know how you are getting to O_K / m and to be honest I am a little unclear about whether this is possible. | |
| Dec 13, 2016 at 8:57 | history | edited | Desiderius Severus | CC BY-SA 3.0 |
Explicit the two other questions and simplifying the form
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| Dec 13, 2016 at 8:26 | history | edited | Desiderius Severus | CC BY-SA 3.0 |
Comments of znt
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| Dec 13, 2016 at 8:21 | comment | added | Desiderius Severus | @znt So for the whole adeles we would get $1 \to \mathbf{C}^\times \times \hat{O}^\times \to \mathbf{A}_K/K^\times \to Cl(K) \to 1$, right ? I simplify my question above for it following your comments. What bothers me the most is: we associate the idele $x$ to the ideal $\mathfrak{a}$ (up to this point, everything is fine), and then reduce modulo $\mathfrak{m}$ and wonder if it lies in the image of $\mathbf{Z}/\mathfrak{m}\cap\mathbf{Z}\to \mathcal{O}_K/\mathfrak{m}$ or not, and here I get totally lost. | |
| Dec 12, 2016 at 16:44 | comment | added | znt | No -- the kernel will be the global units and the cokernel the class group (at least if you meant to use the finite adeles on the right like you did with the rest of your post). Isn't class field theory wonderful! All these fundamental things linked together via this adelic stuff. | |
| Dec 12, 2016 at 16:00 | comment | added | Desiderius Severus | @znt So could it be true without the $K^\times$ ? That is to say, do we have $\hat{O_K}$ isomorphic to $A_K^\times/K^\times$ ? | |
| Dec 12, 2016 at 15:57 | comment | added | znt | What does O_K-hat-star mean? If it means what I think it means (a profinite completion of O_K, and then take the units) then this does not naturally contain K^* so I think some of what you write does not make sense. | |
| Dec 12, 2016 at 13:46 | history | asked | Desiderius Severus | CC BY-SA 3.0 |