## University of Chicago Number Theory Seminar, 2022--2023.

Organizers: Matthew Emerton, Andrea Dotto, Zijian Yao.

Date and time: The seminar meets on Thursday afternoons from 14:00 to 15:30.

Room: Eckhart 308. Talks will be in-person unless stated otherwise.

Mailing list: please email chicago_nt+subscribe@googlegroups.com and follow the instructions there to join the mailing list for this seminar, which is maintained by Zijian Yao.

### Fall 2022 schedule.

10/13/2022: Amina Abdurrahman (Stony Brook).
When do symplectic L-functions have square roots?
We give a purely topological formula for the square class of the central value of a symplectic representation on a curve. This is related to the theory of epsilon factors in number theory and Meyer's signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

10/20/2022: Lue Pan (Princeton).
Regular de Rham Galois representations in the completed cohomology of modular curves.
Let p be a prime. I want to explain how to use the geometry of modular curves at infinite level and the Hodge–Tate period map to study regular de Rham p-adic Galois representations appearing in the p-adically completed cohomology of modular curves. We will show that these Galois representations up to twists come from modular forms and give a geometric description of the locally analytic representations of GL2(Qp) associated to them. These results were previously known by totally different methods.

10/27/2022: Daniel Li-Huerta (Harvard).
The plectic conjecture over local fields.
The étale cohomology of varieties over Q enjoys a Galois action. In the case of Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higher-rank Euler systems, they conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analog of this conjecture for local Shimura varieties. Consequently, we obtain results for the basic locus of global Shimura varieties, after restricting to a decomposition group. The proof crucially uses a mixed-characteristic version of fusion due to Fargues-Scholze.

11/03/2022: Gal Porat (UChicago).
Overconvergence of étale (\varphi,\Gamma)-modules in families.
In recent years, there has been growing interest in realizing the collection of Langlands parameters in various settings as a moduli space with a geometric structure. In particular, in the p-adic Langlands program, this space should come in two different forms of moduli spaces of (\varphi,\Gamma)-modules: there is the "Banach" stack (also called the Emerton-Gee stack), and the "analytic" stack. In this talk, I will present a proof of a recent conjecture of Emerton, Gee and Hellmann concerning the overconvergence of étale (\varphi,\Gamma)-modules in families, which gives a link between the two different moduli spaces.

11/10/2022: Haoyang Guo (Max Planck).
A prismatic approach to Fontaine's C_crys conjecture.
Given a smooth proper scheme over a p-adic ring of integers, Fontaine's C_crys conjecture states that the \'etale cohomology of its generic fiber is isomorphic to the crystalline cohomology of its special fiber, after base changing them to the crystalline period ring. In this talk, we give a prismatic proof of the conjecture, for general coefficients, in the relative setting, and allowing ramified base rings. This is a joint work with Emanuel Reinecke.

11/17/2022: TBA

11/24/2022: Thanksgiving break.

12/01/2022: Andrea Dotto (UChicago).
Multiplicity one and Breuil--Kisin cohomology of Shimura curves.
The multiplicity of Hecke eigenspaces in the mod p cohomology of Shimura curves is a classical invariant which has been computed in significant generality when the group splits at p. These results have recently found interesting applications to the mod p Langlands correspondence for GL_2 over unramified p-adic fields. As a first step towards extending these to nonsplit quaternion algebras, we prove a new multiplicity one theorem in the nonsplit case. The main idea of the proof is to use the Breuil--Kisin module associated to a finite flat model of the cohomology to reduce the problem to a known statement about modular forms on totally definite quaternion algebras.

12/08/2022: TBA

The schedule for the 2021–2022 academic year is here.