Timeline for How to think of algebraic geometry in characteristic p?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jul 30 at 10:06 | history | edited | Martin Sleziak |
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| Jun 30 at 15:32 | answer | added | Will Sawin | timeline score: 9 | |
| Jun 24 at 16:58 | comment | added | Will Sawin | @LSpice Every pseudo-finite field is quasi-algebraically closed, but not vice versa, since e.g. algebraically-closed fields are quasi-algebraically-closed but not pseudo-finite. | |
| Jun 24 at 13:54 | answer | added | jack2045 | timeline score: 4 | |
| S Sep 11, 2023 at 4:07 | history | bounty ended | CommunityBot | ||
| S Sep 11, 2023 at 4:07 | history | notice removed | CommunityBot | ||
| Sep 3, 2023 at 3:54 | comment | added | LSpice | @JasonStarr, re, what is the difference between "quasi algebraically closed" and "pseudo-finite"? | |
| S Sep 3, 2023 at 2:53 | history | bounty started | JustLikeNumberTheory | ||
| S Sep 3, 2023 at 2:53 | history | notice added | JustLikeNumberTheory | Draw attention | |
| Sep 1, 2023 at 10:09 | comment | added | Jason Starr | It is not an interpretation; it is just an analogy. The absolute Galois group of $\mathbb{F}_p$ equals the absolute Galois group of $\mathbb{C}((t))$. Also, both fields are "quasi algebraically closed" fields. | |
| Aug 31, 2023 at 20:34 | comment | added | JustLikeNumberTheory | @JasonStarr I'm primarily interested in just visualizing $\mathbb{A}^1$, along with any other varieties over a field of characteristic $p$. Though admittedly I hope that such a visualization would help to understand theorems in positive characteristic. Could you elaborate on the analogy you laid out? Or point me toward a reference which discusses this analogy in detail; I've never heard of this interpretation. | |
| Aug 31, 2023 at 11:17 | comment | added | Jason Starr | Are you asking for the best visual image of the geometric space of $\mathbb{A}^1$ in characteristic $p$? Or are you asking how we use special features of algebraic geometry in positive characteristic to help us prove theorems? In many ways a closer "geometric analogy" of $\mathbb{A}^n$ over a field like $\overline{\mathbb{F}}_p$ is a fibration over a punctured complex disk where the fibers are isomorphic to affine space. Frobenius is analogous to analytic continuation along a (fraction of a) loop in the disk. | |
| Aug 31, 2023 at 2:06 | comment | added | Donu Arapura | I tend to think of characteristic $p$ algebraic geometry as both similar and different to the complex algebraic geometry, but this probably doesn't answer your question...Turning to the less soft question, supersingular elliptic curves have endomorphism rings much larger than elliptic curves over $\mathbb{C}$, so I don't see how to make an equivalence of categories along the lines you're suggesting. | |
| Aug 30, 2023 at 22:43 | history | asked | JustLikeNumberTheory | CC BY-SA 4.0 |