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

Organizers: Matthew Emerton, Andrea Dotto, Zijian Yao.

Date and time (new for Winter 2023): The seminar meets on Thursday afternoons from 15:30 to 17:00.

Room (new for Winter 2023): Eckhart 207A. Talks will be in-person unless stated otherwise.

Zoom link: https://uchicago.zoom.us/j/97235467744?pwd=RnhQTGRFVklTeEd1cXZwa1ZISzA0Zz09.

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.

Spring 2023 schedule.

03/30/2023: Bogdan Zavyalov (IAS).
Mod-p Poincare Duality in p-adic Analytic Geometry
Etale cohomology of F_p-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the F_p-cohomology of the 1-dimensional closed unit ball is infinite. However, it turns out that the situation is much better if one considers only proper rigid-analytic spaces. These spaces have finite F_p cohomology groups and these groups satisfy Poincare Duality if X is smooth and proper. I will explain how one can prove such results using the concept of almost coherent sheaves that allows to "localize" such questions in an appropriate sense and actually reduce to some local computations.

04/13/2023: David Ben-Zvi (UT Austin).
Relative Langlands Duality
I'll report on joint work with Yiannis Sakellaridis and Akshay Venkatesh, proposing a duality underlying the relationship between L-functions and periods of automorphic forms. The duality is inspired by work of Gaiotto-Witten on electric-magnetic duality for boundary conditions in supersymmetric gauge theory. We view both L-functions and periods as aspects of a "higher" form of geometric quantization applied to suitable Hamiltonian actions of Langlands dual groups. This leads to a series of Langlands duality conjectures (local / global, geometric / arithmetic) which we make precise in the unramified setting over function fields.

04/20/2023: Mathilde Gerbelli-Gauthier (McGill).
Fourier Interpolation and the Weil Representation
In 2017, Radchenko-Viazovska proved a remarkable interpolation result for even Schwartz functions on the real line: such a function is entirely determined by its values and those of its Fourier transform at square roots of integers. We give a new proof of this result, exploiting the fact that Schwartz functions are the underlying vector space of the Weil representation ūĚĎä. This allows us to deduce the interpolation result from the computation of the cohomology of a certain congruence subgroup of ūĚĎÜūĚźŅ2(‚Ą§) with values in ūĚĎä. This is joint work in progress with Akshay Venkatesh.

Winter 2023 schedule.

01/19/2023: Dmitry Kubrak (IHES).
p-adic Hodge theory for Artin stacks.
I will talk about my joint works with Artem Prikhodko, where we try to establish a satisfactory version of p-adic Hodge theory for smooth Artin stacks. I will discuss the older work (https://arxiv.org/abs/2105.05319) where an integral version was proved for certain class of quotient stacks, and also a more recent one (https://arxiv.org/abs/2211.17227) where a rational version was established for all Artin stacks with a smooth Hodge-proper integral model. In the latter work we also develop a truncated version of p-adic Hodge theory in a more general setting of d-Hodge-proper stacks; when applied to schemes it leads to certain purity-type statements for the cohomology of the Raynaud generic fiber and cristallinity of etale cohomology groups in a certain range in the presence of a Cohen-Macauley integral model. If time permits I will also discuss some representation-theoretic applications (which is work in progress with Federico Scavia).

01/26/2023: Eugen Hellmann (Münster). Time change: this talk will be at 09:30am.
This will be a Zoom talk at https://uchicago.zoom.us/j/97235467744?pwd=RnhQTGRFVklTeEd1cXZwa1ZISzA0Zz09.
Towards overconvergent p-adic automorphic forms from a categorical p-adic Langlands view.
It is known - starting with the work of Coleman and Mazur on the eigencurve - that the dual space of (finite slope) overconvergent p-adic automorphic forms can be identified with the global section of a coherent sheaf on a rigid analytic variety, called an eigenvariety. In this talk I will give a conjectural Galois theoretic description of this coherent sheaf in terms of an envisioned categorical p-adic Langlands correspondence (that is inspired by ideas from the geometric Langlands program as well as from the Taylor-Wiles patching method). In the case of GL_2 I will discuss some partial results of this conjectural description.

02/09/2023: Daniel Le (Purdue).
Mod p algebraic modular forms on U(3) at first congruence level at p.
Let F/F^+ be a CM extension and U(n) be a definite unitary group over F^+ which splits over F. The space S of mod p algebraic modular forms is expected to realize a mod p Langlands correspondence satisfying a hypothetical local-global compatibility. One feature of the local mod p correspondence that distinguishes it from the characteristic 0 one is that it involves reducible G = GL_n(F^+ \otimes_Q Q_p)-representations. For n=2, Breuil and Paskunas conjectured that the lattice of G-subrepresentations in S mirrored the lattice of subrepresentations of the local Galois group at p in a precise way based on the invariants of the first principal congruence subgroup at p. Their conjectures have recently been generalized to arbitrary n by Breuil, Herzig, Hu, Morra, and Schraen (BHHMS). We prove the following local-global compatibility result. For n=3, we describe the space of mod p algebraic modular forms at first congruence level at p in terms of the local Galois representation at p and give evidence for the BHHMS conjecture. Our methods include some new techniques to analyze Taylor--Wiles patched modules which are not free over the relevant Galois deformation rings. This is joint work with Bao Le Hung and Stefano Morra.

02/16/2023: Zhongyipan Lin (Northwestern).
A Deligne-Lusztig type correspondence for tame p-adic groups.
Let G be a quasi-split tame p-adic group. In this talk, I will use the Bruhat-Tits building to classify semisimple mod p Langlands parameters and tame inertial types. I will show that the category of characters of Iwahori of maximal unramified tori up to stable conjugacy is equivalent to the category of tame inertial types, which can be thought of as a generalization of the correspondence between Frobenius-stable semisimple conjugacy classes and geometric conjugacy classes of characters studied by Deligne-Lusztig in the finite field case. These discoveries allow us to prove that the Emerton-Gee stacks for all tame groups are noetherian algebraic, and to speculate Serre weight conjectures for quasi-split tame groups.

02/23/2023: Alexander Petrov (Max Planck).
Decomposability of the de Rham complex in positive characteristic.
Deligne and Illusie proved that if a smooth variety over F_p admits a lift over Z/p^2 then the truncation of its de Rham complex in degrees <p is quasi-isomorphic to the direct sum of its cohomology sheaves. As a consequence, the Hodge-to-de Rham spectral sequence of a smooth proper liftable variety degenerates, provided that the dimension of the variety is <=p. However, further truncations of the de Rham complex of a liftable variety need not be decomposable. I will describe the obstruction to decomposing the truncation of the de Rham complex in degrees <=p in terms of other invariants of the variety, and will use this to give an example of a smooth projective variety over F_p that lifts to Z_p but whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. The proof relies on the existence of prismatic cohomology, but the key argument is a computation in homotopical algebra, motivated by a construction of Steenrod operations on cohomology of topological spaces. On the positive side, it turns out that for any smooth variety the de Rham complex becomes decomposable after pulling back along the Frobenius morphism. In particular, the de Rham complex of a Frobenius-split variety is decomposable in all degrees.

03/02/2023: Giovanni Rosso (Concordia).
Overconvergent Eichler‚ÄďShimura morphism for families of Siegel modular forms.
Classical results of Eichler and Shimura decompose the cohomology of certain local systems on the modular curve in terms of holomorphic and anti-holomorphic modular forms. A similar result has been proved by Faltings for the etale cohomology of the modular curve and Faltings' result has been partly generalised to Coleman families by Andreatta‚ÄďIovita‚ÄďStevens. In this talk, based on joint work with Hansheng Diao and Ju-Feng Wu, I will explain how one constructs a morphism from the overconvergent cohomology of GSp_2g to the space of families of Siegel modular forms. This can be seen as a first step in an Eichler--Shimura decomposition for overconvergent cohomology and involves a new definition of the sheaf of overconvergent Siegel modular forms using the Hodge‚ÄďTate map at infinite level. If time allows it, I'll explain how one can hope to use higher Coleman theory to find a complete analogue of the classical Eichler‚ÄďShimura decomposition in small slope.

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‚ÄďTate 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 √©tale cohomology of varieties over Q enjoys a Galois action. In the case of Hilbert modular varieties, Nekov√°Ňô-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.

11/24/2022: 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: Marco D'Addezio (Jussieu).
This will be a Zoom talk. The meeting ID is 919 3477 1507, and the password is the smallest 3-digit prime.
Crew's parabolicity conjecture.
I will report on recent developments of the theory of monodromy groups of convergent and overconvergent F-isocrystals. These algebraic groups have been defined by Crew in '92 using the Tannakian formalism. After quickly recalling the definitions I will explain the main ideas of the proof of Crew's parabolicity conjecture, which provides a strong relationship between these two types of monodromy groups. I will continue by explaining how this comparison can be used as a bridge between "motives" and F-isocrystals/ p-divisible groups. In the end, I will also state an enhancement of the conjecture I recently obtained in a joint work with Van Hoften.

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