On arithmetic characterization of local systems of geometric origin

Abstract: I will talk about the problem of classifying local systems of geometric origin on algebraic varieties over complex numbers, from the point of view of arithmetic.

Conjecture: For a smooth algebraic variety S over a finitely generated field F, a semi-simple Q_l-local system on S_{\bar{F}} is of geometric origin if and only if it extends to a local system on S_{F'} for a finite extension F'\supset F

My main goal will be to provide motivation for this conjecture arising from the properties of the p-adic Riemann-Hilbert correspondence and the Fontaine-Mazur conjecture, and survey known partial results.