Timeline for Computing the genus of certain ternary indefinite lattices
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| May 3, 2021 at 23:45 | history | edited | GH from MO |
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| May 3, 2021 at 21:08 | history | edited | GH from MO |
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| May 3, 2021 at 21:03 | answer | added | Simon Brandhorst | timeline score: 4 | |
| Apr 12, 2021 at 6:33 | comment | added | X77 Math19 | @WillJagy : thank you so much! I will study these references | |
| Apr 11, 2021 at 15:28 | comment | added | Will Jagy | It becomes easier when the ternary is a unary plus a binary. $x^2 + 100 y^2$ and $4x^2 + 25 y^2$ are in the same genus; the other genus is the pair $8x^2 \pm 4xy +13 y^2$ Furthermore, $4x^2 + 25 y^2$ is a square but not a fourth power in the class group, therefore in a different spinor genus from the identity. zakuski.math.utsa.edu/~kap/Estes_Pall_1973.pdf and, really, everything else I put at zakuski.math.utsa.edu/~kap | |
| Apr 11, 2021 at 12:41 | comment | added | X77 Math19 | @WillJagy In your example, $4x^2+25y^2-5z^2=1$ has no solution, since $n=5$ is not congruent to $1,3,7 mod 8$. Reading the introduction of the paper, I may suppose that locally there is a solution (?), but the Hasse principle do not apply, thus this implies that the genus has two elements at least. I see that kind of result in Corollary 1 p.4 of that paper: wordpress.jonhanke.com/wp-content/uploads/2013/04/… however this is rather vague and there is no reference. Do you know such references ? Thank you very much | |
| Apr 11, 2021 at 12:23 | comment | added | X77 Math19 | @WillJagy: thank you so much for the example and the reference! | |
| Apr 10, 2021 at 15:53 | comment | added | Will Jagy | infinitely many examples in example 1.2 in math.uni-sb.de/ag/schulze/Preprints/feixu_rsp_spinorgenera.pdf for instance $4x^2 +25 y^2 - 5 z^2$ | |
| Apr 10, 2021 at 15:45 | comment | added | Will Jagy | the example with 5 and 25 is indefinite forms. Your (indefinite) pattern does not seem to be cooperating as far as producing more than one class in a genus. | |
| Apr 10, 2021 at 5:33 | comment | added | X77 Math19 | @WillJagy I am interested by these forms because they have an order 3 automorphism preserving them. Thank you very much for the examples ; do you know examples that are indefinite ? | |
| Apr 9, 2021 at 22:49 | comment | added | Will Jagy | same for the positive forms $x^2 + xy + y^2 + 9 z^2$ and $x^2 + 3 y^2 + 3yz + 3z^2,$ same genus, each alone in its spinor genus so both are regular in the sense of Dickson. zakuski.utsa.edu/~jagy/papers/Mathematika_1997.pdf | |
| Apr 9, 2021 at 20:38 | comment | added | Will Jagy | oh, well. From Watson's little book, the forms $x^2 +xy-y^2 +25 z^2$ and $5x^2+ 5xy-5y^2 + z^2$ are in the same genus but are distinct. Page 116. | |
| Apr 9, 2021 at 17:29 | comment | added | Will Jagy | why these particular forms? | |
| Apr 9, 2021 at 15:56 | history | edited | X77 Math19 | CC BY-SA 4.0 |
The determinant is $18k'$, not $9k'$ as I wrote
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| Apr 9, 2021 at 13:07 | history | edited | X77 Math19 | CC BY-SA 4.0 |
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| Apr 9, 2021 at 11:34 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
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| Apr 9, 2021 at 11:26 | review | First posts | |||
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| Apr 9, 2021 at 11:23 | history | asked | X77 Math19 | CC BY-SA 4.0 |