Timeline for Does the image of a p-adic Galois representation always lie in a finite extension?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 30, 2021 at 22:53 | history | edited | KConrad | CC BY-SA 4.0 |
added 7 characters in body
|
| Mar 12, 2010 at 17:28 | comment | added | Pete L. Clark | P.S.: I also meant to retain the hypothesis that the base field K is complete (thus doubly ruling out Q_p/Q as a possible counterexample). Otherwise see mathoverflow.net/questions/13346/… | |
| Mar 12, 2010 at 4:22 | comment | added | Pete L. Clark | @KConrad: I lost the word "algebraic": an algebraic normed field extension $L/K$ is complete iff $[L:K]$ is finite. | |
| Mar 12, 2010 at 0:58 | comment | added | KConrad | Something is missing at the end: C_p/Q_p is a normed field extension that is complete but [C_p:Q_p] is not finite (or I could use Q_p/Q instead). Lost a countability hypothesis in the second to last sentence? | |
| Mar 12, 2010 at 0:43 | comment | added | Pete L. Clark | Just a comment: I think the Schikhof proof is very natural from a functional analytic perspective. Indeed, if K is any complete normed field, and V is a normed linear K-space, then it's a basic fact that any finite-dimensional subspace of V is closed (and an obvious fact that any proper subspace has empty interior). It now follows immediately from Baire Category that there is no complete normed K-linear space of countably infinite dimension. Note that, more generally, a normed field extension L/K is complete iff [L:K] is finite. This is proved in Bosch-Guntzer-Remmert. | |
| Mar 12, 2010 at 0:24 | history | edited | KConrad | CC BY-SA 2.5 |
edited body
|
| Mar 12, 2010 at 0:23 | comment | added | KConrad | The switch to the congruence subgroups $G_r$ was actually intentional. At the point in the proof where they start being used, what matters is the way these subgroups shrink nicely to the identity, and for that purpose I thought it was better to emphasize the containment of the terms $g_i$ in the various subgroups $G_{d_i}$ instead of the finer information that they are in the subgroups $K_{d_i}$. However, since the switch admittedly can look like a typo, I added some text to the argument to point out the switch was being (deliberately) made. | |
| Mar 11, 2010 at 23:51 | comment | added | Johnson-Leung | This is the line of argument that I was attempting to make, but I accepted jnewton's answer because it came in first. I wanted to let you know that in your write-up you switch from $K_r$ to $G_r$ midway through the proof. Thanks again! | |
| Mar 11, 2010 at 17:10 | history | edited | KConrad | CC BY-SA 2.5 |
added 2416 characters in body
|
| Mar 11, 2010 at 2:20 | history | edited | KConrad | CC BY-SA 2.5 |
deleted 3701 characters in body; added 2 characters in body
|
| Mar 11, 2010 at 2:12 | history | answered | KConrad | CC BY-SA 2.5 |