Abelian varieties and Selberg class

Hello everyone,

I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions belonging to the Selberg class S in such a way: 1) One associates to a simple abelian variety a primitive function of S, 2) One associates to an abelian variety of dimension d a function of S of degree d, 3) If V is an abelian variety isogenous to a product of abelian varieties of lower dimensions V_1, V_2, ... V_n, then the function F of S related to V is the product of the F_i where F_i is the function of S related to V_i.

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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. –  Pete L. Clark Dec 13 '10 at 19:55
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. –  Sylvain JULIEN Dec 13 '10 at 22:49
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... –  Tim Dokchitser Dec 20 '10 at 1:22
...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. –  Tim Dokchitser Dec 20 '10 at 1:23
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! –  Pete L. Clark Dec 20 '10 at 2:10

(Assuming the abelian varieties are supposed to be defined over $\mathbf{Q}$). For every dimension $d\geq 1$, select a cuspidal automorphic representation $f_d$ of $GL(d)$ over the rationals. Then map any simple $A$ of dimension $d$ to $L(f_d,s)$, and extend by multiplicativity using the simple factors of a general $A$ up to isogeny.
(Of course this question is utterly artificial; it seems reasonable to think that the only natural $L$-function associated to an abelian variety is its Hasse-Weil $L$-function, which has degree $2\dim(A)$.)