Timeline for Finiteness of number of extensions with bounded degree and discriminant
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 6, 2023 at 10:16 | vote | accept | Rami | ||
| Aug 6, 2023 at 10:16 | comment | added | Rami | Thank you very much. We write here the relevant references for the record: 1. [SGA1, Expos´e XII, Th´eor`eme 5.1]. library.msri.org/books/sga/sga/1/1t_332.html 2. stacks.math.columbia.edu/tag/0BND 3. people.math.ethz.ch/~pink/Theses/2018-Bachelor-Noah-Held.pdf Theorem 6.9. | |
| Aug 1, 2023 at 15:56 | comment | added | abx | The cover $X$ of $\mathbb{C}^d\smallsetminus D$ is algebraic, so $K$ is just its field of rational functions, and the extension corresponds to the covering $X\rightarrow \mathbb{C}^d\smallsetminus D$. There is no need for compactification or GAGA argument, just the theory of the algebraic fundamental group. | |
| Aug 1, 2023 at 15:13 | comment | added | Rami | Thank you very much for your answer. We still do not understand one thing: why is the field extension $K/ \mathbb{C}(x_1,\ldots,x_d)$ uniquely determined by the action of $\pi _1(\mathbb{C}^d\smallsetminus D)$ on a set with $n$ elements (in other words, why is the extension uniquely determined by the isomorphism class of the corresponding cover of $\mathbb{C}^d\smallsetminus D$ in the analytic category). Does it follow from some compactification argument or some relative version of GAGA? If so, can you point us to a relevant reference? Thank you very much again. | |
| Jul 18, 2023 at 7:11 | history | answered | abx | CC BY-SA 4.0 |