Geometric Langlands Seminar, 2021--2022

This is the website for the 2021--2022 Geometric Langlands Seminar at UChicago.

Please email me at amathew (at) math (dot) uchicago (dot) edu for the Zoom coordinates. The lecture videos work best if they are first downloaded (or they sometimes stop after an hour).

During March 2022, I gave a series of talks. The relevant article is available here.

During February 2022, Benjamin Antieau gave a series of talks.

During February 2022, I gave a series of pretalks about cyclotomic spectra.

During January 2022, Arthur--César Le Bras gave a series of lectures.

During November 2021, Bhargav Bhatt gave a series of lectures.

During October-November 2021, Alexander Beilinson gave a series of lectures. The introduction to Beilinson's article is available.

- Lecture 1 (10/28): Notes and video.
- Lecture 2 (11/04) Notes and video. The referenced article of Illusie.
- Lecture 3 (11/11) Notes and video

During October 2021, Alexander Petrov gave a series of lectures over Zoom in the Geometric Langlands Seminar at UChicago.

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}}).