Timeline for How small may the discriminant of an $S_d$-field be?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2019 at 3:19 | comment | added | Vesselin Dimitrov | You are right. Take $S = \emptyset$: I should have simply said $\mathrm{Spec} \, O_K$. | |
| Feb 13, 2019 at 18:46 | comment | added | Aurel | @VesselinDimitrov That was my understanding of your notation, but I don't see why a family of $S_d$-extensions of $K$ unramified outside a fixed $S$ should have bounded root discriminant. | |
| Feb 13, 2019 at 18:19 | comment | added | Vesselin Dimitrov | @Aurel: Sorry, due to the word count limitation of the comment window I omitted to state that $S$ is a finite set of places of some number field $K$, and that $O_{K,S}$ is here the ring of $S$-integers of $K$ (not a localization). You are right that my notation was ambiguous at best. | |
| Feb 13, 2019 at 16:34 | comment | added | Aurel | @VesselinDimitrov Why is this enough? There exist extensions of $\mathbb{Q}_p$ with unbounded root discriminant. | |
| Feb 13, 2019 at 10:43 | comment | added | Vesselin Dimitrov | @TimDokchitser: Thank you, in these results though they assume that the degree $d$ is fixed, and don't bother with writing down implied constants depending on $d$. Presumably the growing degrees and small discriminate case is not as well understood. On the other hand, the question would be solved if there existed an $\mathrm{Spec} \, O_{K,S}$ whose etale fundamental group (group of the maximal unramified outside $S$ extension of $K$) is complicated enough to contain an $S_d$-quotient for each $d = 1, 2, \ldots$. | |
| Feb 13, 2019 at 8:37 | comment | added | Tim Dokchitser | Not sure this is in the direction you want, but I think Bhargava-Shankar-Wang's paper "Squarefree values of polynomial discriminants I", arxiv.org/abs/1611.09806 proves a lower bound for the number of $S_d$ number fields of discriminant $<X$ that they conjecture to be essentially sharp. (And they discuss various related results.) | |
| Feb 12, 2019 at 20:50 | history | edited | Vesselin Dimitrov | CC BY-SA 4.0 |
added 388 characters in body
|
| Feb 12, 2019 at 16:28 | history | asked | Vesselin Dimitrov | CC BY-SA 4.0 |