On Local Systems of Geometric Origin

I will discuss the following conjecture: an irreducible Q¯ ℓ-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over some descent of S to a finitely-generated subfield of complex numbers.  I will describe the motivation for this conjecture coming from relative p-adic Hodge theory, known partial results, and possible approaches (not very successful so far) to formulating a purely p-adic (and thus hopefully more tractable) version of this conjecture.  A large part of the talk will be expository, including material based on the ideas of Hélène Esnault, Raju Krishnamoorthy, and Josh Lam.