Abelian varieties and Selberg class

2010-12-13T17:39:53Z

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.

Thank you in advance.

Answer by Denis Chaperon de Lauzières for Abelian varieties and Selberg class

2011-03-13T21:02:38Z

(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 $2\dim(A)$.)

Abelian varieties and Selberg class - MathOverflow

Abelian varieties and Selberg class

Answer by Denis Chaperon de Lauzières for Abelian varieties and Selberg class