Take the 2-minute tour ×  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.  share|improve this
question 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 v