Timeline for On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$
Current License: CC BY-SA 3.0
15 events
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| May 20, 2016 at 8:39 | vote | accept | Serendipity | ||
| May 13, 2016 at 21:04 | comment | added | znt | For weight 1 modular forms you can look at people.maths.ox.ac.uk/lauder/weight1 | |
| May 12, 2016 at 14:06 | history | edited | Serendipity | CC BY-SA 3.0 |
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| May 12, 2016 at 13:24 | answer | added | Jeremy Rouse | timeline score: 16 | |
| May 12, 2016 at 13:04 | comment | added | Dror Speiser | As for more examples, LMFDB has a list of Artin representations, in which those with group equal to $S_3$ and trace of complex conjugation equal to 0 are all modular of weight 1 by the same method of proof, and satisfy a similar solution count property. In fact, all Artin reps. of dimension 2 with trace of complex conjugation equal to 0 are modular, and listed in the LMFDB. This follows from the Langlands-Tunnel theorem for "small" galois extensions, and from some more recent theorems that I don't know to whom to attribute to. (most of these by far will not have an eta product expansion) | |
| May 12, 2016 at 12:47 | comment | added | Dror Speiser | That's your second question, I was more focused on the first ;) | |
| May 12, 2016 at 12:45 | comment | added | Serendipity | @DrorSpeiser Thanks. I was wondering if there can be a more direct proof, without using the dimension / basis of the relevant space of moduli forms of some weight and level and nebentypus. | |
| May 12, 2016 at 12:42 | comment | added | Dror Speiser | Small note: the same problem only with $x^3-x-1$ and $\eta(z)\eta(23z)$ is much more common in the literature, including on mathoverflow, so consider looking for that. Second, I think the level is 108 - the discriminant of the field. Third, the proofs using modular forms are pretty nice: use Hecke's results to prove that the polynomial "is modular", prove the eta product is modular, prove they are equal. | |
| S May 12, 2016 at 10:34 | history | suggested | Aurel | CC BY-SA 3.0 |
added link so it is clear what the OP is talking about.
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| May 12, 2016 at 10:29 | review | Suggested edits | |||
| S May 12, 2016 at 10:34 | |||||
| May 12, 2016 at 10:07 | history | edited | Serendipity | CC BY-SA 3.0 |
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| May 12, 2016 at 9:54 | history | edited | Serendipity | CC BY-SA 3.0 |
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| May 12, 2016 at 9:46 | history | edited | Serendipity | CC BY-SA 3.0 |
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| May 12, 2016 at 9:44 | review | First posts | |||
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| May 12, 2016 at 9:40 | history | asked | Serendipity | CC BY-SA 3.0 |