Timeline for Using algebraic geometry to understand class field theory
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 7, 2021 at 13:26 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 376 characters in body
|
| Apr 6, 2021 at 20:30 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 4 characters in body
|
| Apr 6, 2021 at 18:33 | comment | added | Will Sawin | Yes, this is what's needed. I don't actually know anywhere that the computation is done in that language, but I believe it is possible | |
| Apr 6, 2021 at 17:18 | comment | added | David E Speyer | I never thought before about what happens to (1) in the ramified case. Thinking now: It looks like the fibers of Sym(X) to GeneralizedJacobian(X) are not projective spaces, but projective spaces with a locus deleted which, after extending the base field, looks like a bunch of hyperplanes. I imagine you have to do some sort of computation to show that, if the conductor is chosen large enough, the monodromy around those hyperplane vanishes? | |
| Apr 6, 2021 at 16:42 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 19 characters in body
|
| Apr 6, 2021 at 16:36 | comment | added | David E Speyer | Thanks! A computation that I remember doing when I read Serre was working out the ray class fields of $\mathbb{P}^1$ with respect to the conductors $(0)+(\infty)$ and $2 (\infty)$ from the generalized Jacobian definition: It was really fun to see the equations $y^{p-1}=x$ and $z^p-z = x$ pop out of the abstraction. | |
| Apr 6, 2021 at 15:40 | comment | added | Will Sawin | Excellent explanation! A brief note: (2) works, without any real additional difficulty, for ramified class field theory - just use the "generalized Jacobian" parameterizing line bundles with a trivialization over a fixed divisor $N$. For (1), proving the ramified analogue is much messier. | |
| Apr 6, 2021 at 15:38 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 14 characters in body
|
| Apr 6, 2021 at 15:19 | history | answered | David E Speyer | CC BY-SA 4.0 |