Timeline for "Cute" applications of the étale fundamental group
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| May 2, 2021 at 21:35 | vote | accept | Libli | ||
| Mar 12, 2021 at 23:35 | comment | added | Asvin | For a positive result though, it is known that the truth of a section conjecture implies an algorithm to determine if a curve has a rational point. See the answer to mathoverflow.net/questions/179664/… | |
| Mar 12, 2021 at 23:34 | comment | added | Asvin | I am not sure and by all accounts, it looks to be a really hard conjecture. Until very recently (?), the only cases where we knew the conjecture was when there were no rational points to begin with and then one could prove there were no sections so maybe it's not really an exact fit to your question. Something else in a similar vein is the en.wikipedia.org/wiki/Neukirch%E2%80%93Uchida_theorem | |
| Mar 12, 2021 at 23:11 | comment | added | Libli | @Asvin : interesting. In some concrete examples, is it any easier to find some splitting of these exact sequences than the corresponding rational point? Or at least, is there a way to prove existence of such splittings, again in some concrete examples, without knowing they correspond to rational points in the first place? | |
| Mar 12, 2021 at 18:14 | comment | added | Asvin | If you are happy with conjectures, Grothendieck's section conjecture is another nice candidate. It says that the rational points of hyperbolic curves should be in bijection with splittings of a fundamental short exact sequence involving fundamental groups. | |
| Mar 10, 2021 at 18:41 | answer | added | Asvin | timeline score: 14 | |
| Mar 10, 2021 at 15:52 | comment | added | Asvin | But really, the tate module is so pervasive in arithmetic geometry that pretty much any big result about abelian varieties/jacobians will feature it prominently (Mazur's theorem, Herbrand-Ribet,Faltings'...) but maybe these are too sophisticated? A more elementary example might be the theorem about good reduction - an elliptic curve has good redn at a prime iff the galois action on the tate module is unramified at that prime! | |
| Mar 10, 2021 at 15:49 | comment | added | Asvin | One thing that wowed me when I first saw it was that you could bound the size of the endomorphism ring of your elliptic curve by studying it's action on the Tate module. You could also try doing this with the tangent space but mod p, the tangent space is defined over a finite field so the endomorphisms can't act faithfully but the Tate module works! | |
| Mar 10, 2021 at 14:53 | comment | added | Libli | @Asvin : why not! If I can easily make the link between étale maps and the Tate modules. I must admit I am more inclined to think about complex geometry and I am starting to think again about all these incarnations of the (étale) fundamental group for this course I am teaching. But a cute and easily explainable application of the Tate module to obstruction of rational solutions would certainly be a fact I will be delighted to teach and explain | |
| Mar 10, 2021 at 14:44 | comment | added | Asvin | For many familiar objects (Abelian varieties...) the etale fundamental group can be identified with a more classical, familiar invariant (the Tate module...). Would you count applications of the Tate module for instance, as applications of the etale fundamental group? | |
| Mar 10, 2021 at 14:23 | answer | added | Gabriel | timeline score: 8 | |
| Mar 10, 2021 at 13:25 | comment | added | D.-C. Cisinski | @Libli Chapter 4 on Legendre symbols is nice: they are identified with the analogue of linking numbers (Prop. 4.4). You have to read the part on classical reading number to appreciate it though. That is not an application per se (hence I did not post that an answer), but I find this cute! | |
| Mar 10, 2021 at 11:37 | comment | added | Libli | @Denis-CharlesCisinski : Thanks a lot. I wasn't aware of this book, neither of the theory linking knots and primes. I must admit I am little bit lazy to read the whole book. Do you suggest a specific chapter that would answer to my question? | |
| Mar 10, 2021 at 9:53 | history | edited | Hollis Williams | CC BY-SA 4.0 |
edited title
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| Mar 10, 2021 at 9:05 | comment | added | D.-C. Cisinski | Did you try Morishita's book on analogies between knots and pimes? springer.com/de/book/… A shorter available version is on arXiv: arxiv.org/abs/0904.3399 | |
| Mar 10, 2021 at 9:01 | history | edited | Glorfindel | CC BY-SA 4.0 |
added 2 characters in body
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| Mar 10, 2021 at 8:39 | history | became hot network question | |||
| Mar 10, 2021 at 7:37 | history | edited | Libli | CC BY-SA 4.0 |
edited scope of the examples I am looking for
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| Mar 10, 2021 at 2:12 | comment | added | M.G. | I would argue that your example is rather about the power of functoriality than the fundamental group itself as the (co)homology functor seems better suited when $n>2$. As Brian Conrad puts it, "before functoriality, people lived in caves" :-) Other than that, I definitely share the WOW-sentiment :-) | |
| Mar 10, 2021 at 1:37 | answer | added | user175608 | timeline score: 38 | |
| Mar 10, 2021 at 0:38 | history | asked | Libli | CC BY-SA 4.0 |