Title:
Automatic de Rham-ness of p-adic local systems and Galois action on the pro-algebraic completion of the fundamental group.
Abstract:
Given an etale Q_p-local system L on a smooth variety X over a number field F,
we might wonder whether it appears as a subquotient of the relative cohomology
of a family of varieties. Necessary conditions for this to be possible include L
being de Rham (in the sense of p-adic Hodge theory) and that L extends to an
integral model of X over some partial ring of integers of F.
It turns out that, while there are plenty of local systems violating these
conditions, for any L it is possible to find an auxiliary local system L' that
satisfies the aforementioned properties and such that the restriction of L to
X_{\bar{F}} embeds into the restriction of L'. In particular, (a relative
version of) the Fontaine-Mazur conjecture implies that any semi-simple local
system on X_{\bar{F}} that can be extended to X in some way, should come from
algebraic geometry. The proof relies on p-adic Simpson and Riemann-Hilbert
correspondences, I'll try to review all the necessary foundational results from
relative p-adic Hodge theory and explain the proof.
This result can be reformulated in terms of the Galois action on the space of functions on the pro-algebraic completion of the fundamental group of X_{\bar{F}}: every finite-dimensional Galois subrepresentation of this space is de Rham and almost everywhere unramified. That is, such rings of functions can serve as a source of Galois representations satisfying the assumptions of the Fontaine-Mazur conjecture. I'll then try to explain that this source is in a certain sense universal: for X the projective line with 3 punctures any semi-simple Galois representation coming from etale cohomology of an algebraic variety can be established as a subquotient of the space of functions on the pro-algebraic completion of pi_1^et(X_{\bar{F}}).