Galois action on the pro-algebraic fundamental group

Abstract: 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 an almost everywhere unramifed.

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.

algebraic geometrynumber theory

Audience: researchers in the topic

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MIT number theory seminar

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Past semesters

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