# Galois action on the pro-algebraic completion of the fundamental group

SEMINARS, NUMBER THEORY

##### Speaker:

Alexander Petrov *- Harvard University*

Given a variety over a number field, its geometric etale fundamental group comes equipped with an action of the Galois group. This induces a Galois action on the pro-algebraic completion of the etale fundamental group and hence the ring of functions on that pro-algebraic

completion provides a supply of Galois representations.

It turns out that any finite-dimensional p-adic Galois representation contained in the ring of functions on the pro-algebraic completion of the fundamental group of a smooth variety satisfies the assumptions of the Fontaine-Mazur conjecture: it is de Rham at places above p and is a. e. unramified.

Conversely, we will show that every semi-simple representation of the Galois group of a number field coming from algebraic geometry (that is, appearing as a subquotient of the etale cohomology of an algebraic variety) can be established as a subquotient of the ring of functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.