Timeline for History of "no positive definite ternary integral quadratic form is universal"?
Current License: CC BY-SA 3.0
15 events
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| Feb 13, 2022 at 9:45 | comment | added | Ivan Meir | @PeteL.Clark Please see my answer relating to your comment about an elementary proof. I know this is a long time after your original post and apologies if this was already known to you but thought it worth adding for completeness. | |
| Sep 15, 2011 at 3:04 | comment | added | Will Jagy | As to part regularity on an arithmetic progression, the best example is Ramanujan's form $x^2 + y^2 + 10 z^2,$ which is not regular, but does represent all $6 n + 5$ integrally. First proof by J. S. Hsia (1993 letter to Kap), later elementary proof by me, finally in Oh's A.A. paper. | |
| Sep 15, 2011 at 2:59 | comment | added | Will Jagy | Pretty recent, Byeong-Kweon Oh in Acta Arithmetica (I believe that is where it appeared), proved 8 out of 22, so there are 14 to go. I don't believe I have it on the website, here's him, math.snu.ac.kr/~bkoh and the title is number 24 in math.snu.ac.kr/~bkoh/papers.html while number 28 is the promised application to arithmetic progressions. | |
| Sep 15, 2011 at 2:20 | comment | added | GH from MO | @Will: Thanks, this is good stuff. In the paper you say for 22 in the list regularity has not been verified yet. Has it been done by Bhargava-Hanke or someone else? | |
| Sep 15, 2011 at 0:32 | comment | added | Will Jagy | GH, it's a positive ternary form with integer coefficients that integrally represents all integers that it represents $p$-adically, or, what is the same, that integrally represents all numbers that are integrally represented by some form in its genus. See: zakuski.math.utsa.edu/~kap/Forms/Kap_Jagy_Schiemann_1997.pdf | |
| Sep 14, 2011 at 23:15 | comment | added | GH from MO | @Will: Thanks for the info, I still think that my memory is closer to the truth. What are "regular ternary forms"? Sorry for my ignorance. | |
| Sep 14, 2011 at 22:02 | comment | added | Will Jagy | GH, I was repeating what Manjul said at the memorial conference for Kaplansky. We all gave five or ten minute remembrances. I suppose Manjul may have rewritten history a little for the occasion. He also said, perhaps not entirely accurately, that he had needed to know how many regular ternaries there were, and the next thing he saw a paper by Kap and others with the title "There are 913 regular ternary forms." This got a big laugh. I told the dishwasher story and the one with "Kaplansky Saves." | |
| Sep 14, 2011 at 21:32 | comment | added | GH from MO | Just a quick historical remark: I believe Bhargava found a simple proof of the 15 Theorem in the beginning of his graduate studies, i.e. I would not say he "was looking for diversions from his own dissertation". Actually the way I remember he was a second year student thinking about class groups for his thesis (and not yet about composition laws that constituted his dissertation in the end). | |
| Sep 14, 2011 at 21:31 | comment | added | Will Jagy | I agree. Perhaps the right thing to say is that Gauss, who studied ternaries as a method for studying binaries, probably has a proof that any positive ternary misses an entire arithmetic progression. That is worth mentioning...for an odd prime, if the form represents a quadratic residue, a nonresidue, $p$ times res, $p$ times nonres, then $p$ is not actually involved even if it divides the discriminant. Similar for $p=2,$ the classes on page 43 of Cassels Rational Quadratic Forms. So maybe Gauss says that.. Do you have Dickson's History available, three volumes? | |
| Sep 14, 2011 at 20:52 | comment | added | Pete L. Clark | @Will: thanks. It looks like Conway's argument is actually quite similar to the one I gave above (or, perhaps, the other way around...). But this is a recent text. My guess is that this result predates explicit consideration of $p$-adic numbers: how does one prove it "without leaving $\mathbb{Z}$"? | |
| Sep 14, 2011 at 20:26 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Sep 14, 2011 at 19:55 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Sep 14, 2011 at 19:49 | comment | added | Pete L. Clark | Hi, Will. First, sincere apologies for not contacting you recently. ("It's not you -- it's me.") Second, thanks for your response. I don't own Conway's book (and, honestly, I am not the biggest fan of his expository style): I wonder if you could sketch his proof? Third, about the global relations on the norm residue symbol: I know, that's what I used in my proof above! Your comment can be refined to: any definite ternary form over $\mathbb{Q}$ (or any number field $K$with exactly one real place...) fails to represent an entire $v$-adic square class for some finite place $v$ of $K$. | |
| Sep 14, 2011 at 19:39 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Sep 14, 2011 at 19:23 | history | answered | Will Jagy | CC BY-SA 3.0 |