Algebra and Geometry Seminar
Given a smooth variety X over a number field, the action of the Galois group on the geometric etale fundamental group of X makes the ring of functions on the pro-algebraic completion of this fundamental group into a (usually infinite-dimensional) Galois representation. This Galois representation turns out to satisfy the following two properties:
1) Every finite-dimensional subrepresentation of it satisfies the assumptions of the Fontaine-Mazur conjecture: it is de Rham and almost everywhere unramified.
2) If X is the projective line with three punctures, the semi-simplification of every Galois representation of geometric origin is a subquotient of the ring of regular functions on the pro-algebraic completion of the etale fundamental group of X.
I will also discuss a conjectural characterization of local systems of geometric origin on complex algebraic varieties, arising from property 1) above.