Timeline for Did Gauss know Dirichlet's class number formula in 1801?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 11, 2012 at 20:48 | vote | accept | Jonah Sinick | ||
| Oct 11, 2012 at 17:31 | answer | added | Franz Lemmermeyer | timeline score: 25 | |
| Oct 11, 2012 at 0:29 | comment | added | Henry Cohn | Jonah is right about rings of integers in number fields, but that's a slightly different question (the $h(5^{2k+1})$ example involves non-maximal orders). | |
| Oct 10, 2012 at 23:53 | comment | added | Nikita Sidorov | I'm looking at a paper by Golubeva (springerlink.com/content/y7166487w8q14pp6/fulltext.pdf) where it is mentioned in the introduction that $h(5^{2k+1})=1$ for all $k\ge0$. There is no proof there, but apparently it is well known. So, the question is really about positive density. | |
| Oct 10, 2012 at 23:26 | comment | added | Jonah Sinick | @ Nikita Sidorov: It's not even known that there are infinitely many number fields of any type that have class number 1. What you have in mind might be Hirzebruch and Zagier's paper "Class numbers, continued fractions and the Hilbert modular group" but the paper never appeared and I don't know what's in it. | |
| Oct 10, 2012 at 23:17 | comment | added | Nikita Sidorov | Am I right that it is still not known whether there exists a set of $d>0$ of positive density for which $h_d=1$? I've heard a talk (long time ago) where this was referred to as a "Gauss problem". There is also some nice connection between $h_d$ and the length of the period of the continued fraction expansion of $\sqrt{d}$ but I don't quite remember what it was. | |
| Oct 10, 2012 at 22:10 | comment | added | Jonah Sinick | @ Marty - BTW, Shimura has a great article discussing a common framework for thinking about the ternary quadratic form given by the discriminant and the ternary quadratic form that you mention: ams.org/journals/bull/2006-43-03/S0273-0979-06-01107-4 | |
| Oct 10, 2012 at 22:08 | comment | added | Jonah Sinick | @ Marty - aah, good point, I forgot about that result of Gauss. I wonder if there's an analogous result involving class numbers of real quadratic fields. | |
| Oct 10, 2012 at 21:14 | comment | added | Marty | A short comment for now: Gauss understood the connection between lattice points on spheres and class numbers of definite quadratic forms: The number of representations of $m$ as a sum of 3 squares is a constant times $h(-m)$ or $h(-4m)$ depending on the congruence of $m$ mod $8$, as I recall. The estimate (a) can probably be deduced from counting lattice points in $R^3$. | |
| Oct 10, 2012 at 20:38 | history | asked | Jonah Sinick | CC BY-SA 3.0 |