# 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.

• 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 20, 2010 at 1:22
• 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 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 $2\dim(A)$.)