Timeline for Abelian varieties and Selberg class
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9 events
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| Mar 13, 2011 at 21:02 | answer | added | Denis Chaperon de Lauzières | timeline score: 2 | |
| Dec 20, 2010 at 2:10 | comment | added | Pete L. Clark | Sorry, I know almost nothing about the Selberg Class and was using the wikipedia article on the subject as a reference. But worse than that, I wasn't reading it carefully. My thought was as simple as the following: if you scale an element of the Selberg class by a nonzero real number you get another element of the Selberg class. But actually this is not true: the Dirichlet series needs to be "normalized": $a_1 = 1$. So I retract my comment about uncountability! | |
| Dec 20, 2010 at 1:29 | history | edited | Tim Dokchitser |
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| Dec 20, 2010 at 1:23 | comment | added | Tim Dokchitser | ...are supposed to be in the Selberg class. I am not an expert, but I thought there are motives (e.g. cohomology groups of some Calabi-Yaus?) that are not realizable inside abelian varieties because the weights are wrong. But otherwise that that, I think it is expected that there is a map like this. | |
| Dec 20, 2010 at 1:22 | comment | added | Tim Dokchitser | I think you need to assume the Hasse-Weil conjecture (an L-function of every abelian variety over ${\mathbb Q}$ is entire with the expected functional equation), so get this map; if I am not mistaken, this does not obviously follow just from Selberg orthonormality. Also, as Pete remarks, the image is not the whole Selberg class. I do not know about uncountability, but e.g. you won't get non self-dual $L$-functions, like those of Dirichlet characters of order $\gt 2$. And, even restricting to the self-dual ones, $L$-functions of all etale cohomology groups of all algebraic varieties... | |
| Dec 13, 2010 at 22:49 | comment | added | Sylvain JULIEN | Indeed the word "correspondence" may not fit exactly what I have on my mind, but my English is far from being perfect (I'm French). I didn't know that there were uncountably many functions in S, would you have some reference? By the way, still assuming Selberg's orthonormality conjecture, do both abelian varieties and Selberg's class form semi-simple categories? If so, the concept of functor from abelian varieties into Selberg's class, as you suggested, may be the good way to express my idea. | |
| Dec 13, 2010 at 19:55 | comment | added | Pete L. Clark | This is an interesting question, though I know little about Selberg's class S. One comment though: when you say "natural correspondence between", it makes it sound like you mean a bijection. (That seems unlikely in this case, because there are uncountably many functions in Selberg's class and only countably many abelian varieties over number fields.) But anyway reading more carefully it looks like you are not suggesting a correspondence, but rather a mapping (functor?) from abelian varieties into Selberg's class. | |
| Dec 13, 2010 at 19:29 | history | edited | François G. Dorais |
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| Dec 13, 2010 at 17:39 | history | asked | Sylvain JULIEN | CC BY-SA 2.5 |