Timeline for Intuitive reasons for the existence of modular parametrizations
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 14, 2015 at 18:25 | comment | added | Will Sawin | @მამუკაჯიბლაძე If you want to understand everything about how the local theory relates to the global theory, I think yes. You can express everything with classical modular forms, but some things will be less elegant. | |
| Nov 13, 2015 at 8:53 | comment | added | მამუკა ჯიბლაძე | So do you imply that ultimately to understand all this one has to translate modular forms into the language of automorphic representations? | |
| Nov 12, 2015 at 13:54 | comment | added | Will Sawin | @მამუკაჯიბლაძე There is. From a modular form and a prime $p$ you obtain an irreducible representation of $GL_2(\mathbb Q_p)$. This is from viewing the modular form as a differential form on $\lim_{n \to \infty} X(p^n)$, which has an action of $GL_2(\mathbb Q_p)$, and taking the representation it generates, which turns out to be irreducible. This is the "reduction mod $p$". | |
| Nov 12, 2015 at 6:57 | comment | added | მამუკა ჯიბლაძე | As for your last remark - this is in fact a good example to argue that there are never any pure coincidences. After all, your equality of Lie groups can be explained by $\mathbb C\mathbf P^1=S^2$. You can call also this a coincidence but... | |
| Nov 12, 2015 at 6:54 | comment | added | მამუკა ჯიბლაძე | This is a very good answer, thank you! The key seems to be in the third paragraph, may I ask you to just slightly expand it? I mean, is there any sense in the combination of words "reduction of a modular form at a prime"? | |
| Nov 12, 2015 at 4:49 | history | answered | Will Sawin | CC BY-SA 3.0 |