Algebra/Topology Seminar

Speaker: Alexander Petrov

Title: Galois action on the pro-algebraic fundamental group

Abstract: Given an algebraic variety over a number field F, the algebro-topological invariants of its space of complex points can be equipped (after being profinitely completed) with an action of the Galois group of F. The Galois action on cohomology with coefficients in Q_p is quite well-studied: it is known to be almost everywhere unramified and de Rham at places above p.

I will discuss the analogous result for the fundamental group: for a smooth variety X over F, the Galois action on the pro-algebraic completion of the etale fundamental group of X over \bar{F} is de Rham and locally almost everywhere unramified, in the appropriate sense.

Conversely, it turns out that any semi-simple Galois representation appearing in the cohomology of an algebraic variety over F can be exhibited as a subquotient of the ring of functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures. Assuming the Fontaine-Mazur conjecture, this implies that the class of irreducilbe subquotients of the ring of functions on the pro-algebraic completion of π₁(P¹\{0,1,∞}) is precisely the class of irreducible Galois representations appearing in the cohomology of varieties.