Assume that semi-simplicity of the Galois action on $\ell$-adic cohomology of all smooth projective varieties over finite fields, were known.


Can one deduce that the Galois action on $\ell$-adic cohomology of smooth projective varieties over **number fields** is semi-simple too?



Semi-simple Galois actions on étale cohomology
No. For example, it follows from Weil's results in the 1940s that the Galois action on the etale cohomology of abelian varieties over finite fields is semisimple, but it was not until Faltings in 1983 that one knew the similar result over number fields.


