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.

Title: Syntomic complexes and p-adic étale Tate twists.

Abstract: I will give an introduction to the syntomic complexes defined by Bhatt--Lurie and Bhatt--Morrow--Scholze using prismatic cohomology (generalizing earlier constructions of Fontaine--Messing and Kato). I will explain a description of syntomic complexes for p-torsionfree regular schemes in terms of p-adic vanishing cycles. Joint with Bhargav Bhatt.

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

Title: Cartier modules and cyclotomic spectra.

Abstract: I will describe work with Thomas Nikolaus describing a t-structure on cyclotomic spectra and how to use the new theory of topological Cartier modules to identify the heart with an abelian category of Cartier modules, certain abelian groups with F and V operators satisfying familiar relations from p-typical Witt vectors.
During February 2022, I gave a series of pretalks about cyclotomic spectra.

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

Title: Fourier transform and the geometrization of the local Langlands correspondence.

Abstract: I will explain how to define an $\ell$-adic Fourier transform for Banach-Colmez spaces (what these objects are will be also explained). This construction is motivated by considerations coming from Fargues-Scholze's geometrization program of the local Langlands correspondence, and I will try to illustrate it on examples. This is joint work with Johannes Anschütz, see
During November 2021, Bhargav Bhatt gave a series of lectures.

Title: Absolute prismatic cohomology

Abstract: These talks concern the stacky approach to the absolute prismatic site and its cohomology (developed by Drinfeld and independently in my work with Lurie). This approach helps geometrize the study of prismatic crystals and their cohomology. In these talks, I will describe this approach as well as some concrete outputs including: (a) Sen theory via the Cartier-Witt stack and Drinfeld's refinement of the Deligne--Illusie decomposition, (b) a simple proof of the Hodge-Tate comparison for relative prismatic cohomology, (c) the prismatic logarithm (which leads to a theory of Chern classes), and (d) a fibre square describing the absolute Nygaard filtration in terms of the relative one.

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

Title: Height pairing and vanishing cycles

Abstract: Suppose we have a family of smooth projective varieties over Q with singular fiber having isolated singularities such that the blowup of them is smooth. Spencer Bloch conjectured that the height pairing between algebraic cycles supported on the exceptional divisor of the blowup can be read from the limiting Hodge structure. I will give a proof.

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

Title: Automatic de Rham-ness of p-adic local systems and Galois action on the pro-algebraic completion of the fundamental group.


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}}).