Timeline for Indefinite quadratic form universal over negative integers
Current License: CC BY-SA 3.0
13 events
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| Mar 9, 2017 at 3:41 | comment | added | Will Jagy | @SashaKolpakov see Corollary 3.2 on page 170 of T. Y. Lam, Introduction to Quadratic Forms over Fields. If a rational number $a$ is not represented, it is because there is some $\mathbb Q_p$ over which it is not represented. Since you have an indefinite ternary, $p \neq \infty.$ So there are positive and negative rationals missed, given by local conditions. | |
| Mar 9, 2017 at 3:05 | comment | added | SashaKolpakov | Dear Will, one more question: let a quad form f with rational coefficients miss a rational number. Will it imply that f misses a negative rational number (allowing numbers to be represented rationally, not only integrally). My original question was about forms missing integers rationally. In fact, not representing a negative rational number over Q (for a form over Q) is what I'm looking at. | |
| Feb 28, 2017 at 19:47 | comment | added | SashaKolpakov | Thank you for your help! I'll have a look and try to figure out what I can do: I'm eager to learn more. Sorry about reversing Dickson's statement. Indeed, it was stupid of me! | |
| Feb 28, 2017 at 18:49 | comment | added | Will Jagy | @SashaKolpakov meanwhile, it seems possible to me that the methods of Dickson suffice for this problem. So, unless John Voight takes an interest, I suggest you write to Gonzalo Tornaria, who has been helpful to me. cmat.edu.uy/~tornaria As far as Dickson (you did not state his result correctly) you mostly want his Studies (about 1930) and Modern Elementary (1939) | |
| Feb 28, 2017 at 17:08 | comment | added | Will Jagy | @SashaKolpakov I put a few dozen pdfs at zakuski.math.utsa.edu/~kap For indefinite ternaries, the big result of Eichler was that equivalence class and spinor genus coincide. If spinor genus and genus coincide, the numbers represented are specified entirely by congruences; your condition on negative numbers says there are no congruence obstructions. | |
| Feb 28, 2017 at 10:53 | comment | added | SashaKolpakov | Dear Will, thank you again! I will need however some more information about the link between genera and spinor genera with representability (you consider the case when they don't coincide, but what happens if they do?) I'm sure you may provide me with a reference for further reading which will help understanding things and finishing the "impossibility" proof (I do need to write a proof to this fact and thus need to learn!) Although your help has already been great. Thank you! | |
| Feb 27, 2017 at 11:00 | vote | accept | SashaKolpakov | ||
| Feb 27, 2017 at 4:13 | comment | added | Will Jagy | @SashaKolpakov I do not believe that to be possible. That's what I was talking about in the last paragraph, beginning with "Well, this seems to do it." | |
| Feb 27, 2017 at 3:55 | comment | added | SashaKolpakov | Dear Will, thank you for this explanation. If I may, I would like to ask a more precise question: is there a reference to an example of a ternary quadratic form that represents all negative integers over Q, but avoids some positive integer (I need forms over Q mostly). If no reference exists, is it possible to come up with a concrete example? In my mind representing every negative integer (over Q, not integrally) is quite a strong property to be checked. | |
| Feb 26, 2017 at 20:30 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Feb 26, 2017 at 20:24 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Feb 26, 2017 at 20:01 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Feb 26, 2017 at 19:45 | history | answered | Will Jagy | CC BY-SA 3.0 |