Timeline for Using algebraic geometry to understand class field theory
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2021 at 8:15 | comment | added | Gabriel | @MattE indeed. I think I have a satisfactory answer for the function field case, but I am also interested in the number field case. What I am not yet sure is if we don't have to use class field theory to prove Artin-Verdier. | |
| Mar 29, 2021 at 1:22 | comment | added | Matt E | Dear Gabriel, Maybe ... . But I would guess Lang is more referring to geometric proofs of Class Field Theory in the function field case which rely on simple geometric properties of $\mathbb P^1$, which is something a bit different and more concrete than the sheaf theory of Artin--Verdier. | |
| Mar 28, 2021 at 17:17 | comment | added | Gabriel | Dear Matt, I may be too naïve (and I surely don't yet understand even a small part of this) but isn't this (Artin-Verdier duality) precisely what I wanted? If I recall correctly, we can deduce all of class field theory from Tate's duality. Then if we can prove (a generalization) of Tate's duality using étale cohomology, this would be precisely what I want. | |
| Mar 28, 2021 at 16:24 | comment | added | Squid with Black Bean Sauce | There is a beautiful writeup by Brian Conrad and Alessndro Maria Masullo here: math.stanford.edu/~conrad/BSDseminar/Notes/L4.pdf | |
| Mar 28, 2021 at 13:44 | history | answered | Matt E | CC BY-SA 4.0 |