Timeline for Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 25, 2017 at 10:31 | history | edited | Myshkin | CC BY-SA 3.0 |
+ top level tag (nt.) + ref. req.
|
| Nov 3, 2012 at 13:15 | vote | accept | Keenan Kidwell | ||
| Nov 3, 2012 at 8:37 | comment | added | Olivier | No examples are given, but from his description of the general construction and say the content of Weber Algebra, you get the anticyclotomic extension for free. | |
| Nov 3, 2012 at 8:34 | comment | added | Olivier | I think the idea of $\mathbb Z_{p}$-extension is the kind of idea that have been around at least implicitly for a long time. Certainly Kronecker and Weber knew explicit descriptions of abelian extensions of CM fields, and from that knowledge, introducing the $\mathbb Z_{p}$-extension is just singling out some particularly interesting extensions. I think that Iwasawa is probably the person who first explicitly introduced them. Already in his 1959 article, the general set-up is in place and a general construction of $\mathbb Z_{p}$-extension is given using class field theory. | |
| Nov 3, 2012 at 5:47 | comment | added | Filippo Alberto Edoardo | ...I would say it is the most obvious example beside cyclotomic one, so I guess Iwasawa himself must have thought about it especially when looking for extensions with non-trivial invariants. | |
| Nov 3, 2012 at 5:46 | comment | added | Filippo Alberto Edoardo | I do not really know here it first appears, but I think you should have a look at the 1973 paper by Iwasawa "On the $\mu$ invariants in $\mathbb{Z}_\ell$-extensions". Note that for imaginary quadratic fields, Leopoldt conjecture is almost obvious, as it is for $\mathbb{Q}$, because there are no global units beside roots of units, so it was a standard fact that the compositum of all $\mathbb{Z}_p$-extensions of an imaginary quadratic field K has Galois group $\mathbb{Z}_p^2$, on which $\mathrm{Gal}(K/\mathbb{Q})$ acts, and identifies the anticyclotomic extension as −1-eigenspace... | |
| Nov 3, 2012 at 3:46 | answer | added | Yuri Zarhin | timeline score: 6 | |
| Nov 3, 2012 at 0:28 | history | asked | Keenan Kidwell | CC BY-SA 3.0 |