p-adic Geometry, p-adic Hodge theory, and Shimura varieties

Schedule

Monday, July 28 (Yates 104)

• 1:30 - 2:20      Wiesława Nizioł           p-adic pro-etale cohomology of rigid analytic varieties

• 2:50 - 3:40      Haoyang Guo              A one-line formula of Breuil--Kisin modules and a prismatic Riemann--Hilbert functor

• 4:10 - 5:00      Padma Srinivasan       Mac Lane valuations and arithmetic applications

     Tuesday, July 29 (Yates 104)

• 1:30 - 2:20      Maria Fox                     Traverso's Isogeny Conjecture and Unitary p-Divisible Groups

• 2:50 - 3:40      Naomi Sweeting           On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations

• 4:10 - 5:00      Lucas Mann                  A mod p Lubin-Tate tower and its cohomology 

     Wednesday, July 30 (in Ballroom)

• 1:30 - 2:20      Ananth Shankar            The Andre-Pink-Zannier conjecture in characteristic p

• 2:50 - 3:40      Sean Howe                   p-adic manifold fibrations, inscription, and twistors

• 4:10 - 5:00      Michael Harris               Square root p-adic L-functions

     Thursday, July 31 (Yates 104)

• 1:30 - 2:20      Juan Camargo               D-modules on arc-stacks

• 2:50 - 3:40      Brandon Levin                Modularity lifting in dimension three

• 4:10 - 5:00      Tony Feng                      Symplectic arithmetic duality on Brauer groups

Abstracts

• Wiesława Nizioł

Title: p-adic pro-etale cohomology of rigid analytic varieties

Abstract: The last ten years have seen a a major progress in understanding p-adic pro-etale cohomology of rigid analytic varieties. I will review  briefly key theorems, computations, and examples with a bias towards applications in p-adic Langlands Program. 

• Haoyang Guo

     Title: A one-line formula of Breuil--Kisin modules and a prismatic Riemann--Hilbert functor

     Abstract: In p-adic Hodge theory, a fundamental observation of Breuil and Kisin is that some Galois representations                      over p-adic integers give rise to interesting integral linear-algebraic data, where the latter nowadays are called Breuil--Kisin modules. This association from Galois representations to Breuil--Kisin modules is however very complicated and has so far lacked an explicit description. In this talk, we give a one-line formula for the Breuil--Kisin module of a crystalline or a semi-stable representation. Moreover, by extending this idea, we introduce a prismatic enhancement of the p-adic Riemann--Hilbert functor for p-adic local systems, answering a question of Esnault.

• Padma Srinivasan

Title: Mac Lane valuations and arithmetic applications

Abstract:  In 1936, Mac Lane gave a compact inductive description for geometric valuations on the rational function field (i.e. discrete valuations on the rational function field with residue field of transcendence degree 1), akin to the description of the Gauss valuation. We will describe Mac Lane's theory and survey some recent applications to problems in algorithmic number theory.

• Maria Fox

Title: Traverso's Isogeny Conjecture and Unitary p-Divisible Groups

      Abstract: Given a p-divisible group X, its p-torsion group X[p] and its Newton polygon can both be thought of as classical invariants that capture partial information about X. In some cases, such as when X is the p-divisible group of an elliptic curve, its p-torsion group is enough to determine the Newton polygon. In general, Traverso's Isogeny Conjecture (proven by Nicole and Vasiu in 2007) predicts how much p-power torsion is needed to determine the Newton polygon. We'll discuss how the situation changes when X comes from a point on a unitary Shimura variety. (The new result in this talk is joint with Andrews, Bhamidipati, Goodson, Groen, and Nair.)

• Naomi Sweeting

Title: On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations

Abstract: The Bloch-Kato conjecture is a far-reaching generalization of the famous conjecture of Birch and Swinnerton-Dyer on L functions of elliptic curves. This talk is about recent results towards Bloch-Kato in rank 0 and 1 for spin L-functions of certain automorphic representations of GSp4​. I'll explain the statements and some ideas of the proof, which is based on constructing ramified Galois cohomology classes via level-raising congruences.

• Juan Camargo

Title: D-modules on arc-stacks

Abstract: In this talk, I will discuss a work in progress with Anschütz, Bosco, Le Bras and Scholze concerning a new foundational framework for the theory of the analytic de Rham stack. In particular, we prove that under some mild finite dimensional conditions the formation of the analytic de Rham stack satisfies arc-descent. This allows us to define a very good behave theory of D-modules (from a 6-functor point of view) on a very largeclass of arc-stacks, that notably includes all those that appear in practice in p-adic Hodge theory, and in the geometrization program of Fargues-Scholze.  As an application, by looking at the de Rham stack of Div^1, we give a new proof of the p-adic monodromy theory (more precisely Crew's conjecture) that only uses the theory of Banach-Colmez spaces and the analytic de Rham stack. 

• Michael Harris

    Title:  Square root p-adic L-functions

    Abstract:  Abstract:  The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, gives explicit formulas for central critical values of a large class of Rankin-Selberg tensor products.  The version for unitary groups is now a theorem, and expresses the central critical value of L-functions of the form L(s,∏ x ∏') in terms of squares of automorphic periods on unitary groups.  Here ∏ x ∏' is an automorphic representation of GL(n,F) x GL(n-1,F) that             descends to an automorphic representation of U(V) x U(V'), where V and V' are hermitian spaces over F, with respect to a Galois involution c of F, of dimension n and n-1, respectively.  

    I will report on the construction of a p-adic measure that interpolates the automorphic period — in other words, the square root of the central values of the L-functions — when ∏' varies in a Hida family.  The construction is based on a theory of p-adic differential operators developed by Eischen, Fintzen, Mantovan, and Varma.  This theory has recently been given a new interpretation in terms of p-adic geometry by Graham, Howe, and van Hoften, which has the                     advantage of providing a uniform approach to values of the measure on algebraic characters and on characters of finite order.  In particular, the new approach     provides a geometric explanation for the shape of the local Euler factor at p that was discovered by Yamana and Hsieh, in our forthcoming paper that extends the construction when n = 3 to a 5-variable p-adic L-function.  If time permits, I'll also say something about prospects of generalizing the construction to higher Hida families.

• Brandon Levin

Title: Modularity lifting in dimension three

Abstract: I will discuss work in progress, joint with Robin Bartlett and Bao V. Le Hung, on modularity lifting for three-dimensional Galois representations that are crystalline of minimal regular weight at all places above p. I will begin by reviewing the Taylor-Wiles method and the analogous results in dimension two due to Kisin. I will then outline our approach to generalizing Kisin’s resolution of the local crystalline deformation ring to the three-dimensional setting. Time permitting, I will also discuss potential applications.

• Tony Feng

Title: Symplectic arithmetic duality on Brauer groups

Abstract:  In his 1966 Bourbaki report, Tate conjectured that the Brauer group of a surface over a finite field has a natural symplectic structure. I will motivate the conjecture and explain some elements of its proof. The coup de grâce in this story, which is joint with Shachar Carmeli, should fall between pressing “send” on this e-mail and the SRI. Although the applications are classical, the proof relies on recent advances in perfectoid geometry and prismatic cohomology, especially a theory of prismatic Steenrod operations that we develop.

• Ananth Shankar

     Title: The Andre-Pink Zannier conjecture in characteristic p

     Abstract: The Andre-Pink Zannier conjecture (proved by Richard and Yafaev) addresses the distribution of Hecke orbits in Shimura varieties over number fields. In this talk, I will address a case of this conjecture in positive characteristic, and more generally, I will address the question of how Hecke orbits are distributed in characteristic p. This is joint work with Josh Lam.  

• Lucas Mann

A fundamental result in p-adic geometry is Faltings' observation that infinite-level Lubin-Tate space and Drinfeld space agree, providing a tight connection between these two towers. This observation has many applications to the Langlands program and by a recent observation of Barthel-Schlank-Stapleton-Weinstein it has a surprising application to chromatic homotopy theory: it allows them to compute the rational homotopy groups of the K(n)-local sphere. In the pursuit of extending their computation to include also the p-torsion of these homotopy groups, we introduce a p-torsion variant of the Lubin-Tate tower and the Drinfeld tower and use their cohomology to obtain new results towards the chromatic splitting conjecture. To compute the cohomology, we use our recently developed 6-functor formalism (joint with Anschütz and Le Bras), which should also provide a new way to compute the pro-étale cohomology of the usual Drinfeld space. The project is joint work in progress with a fairly large group of people formed at an AIM workshop last year.

• Sean Howe

    Title: p-adic manifold fibrations, inscription, and twistors 

    Abstract: We will describe some differential structures that arise naturally in the study of moduli spaces in p-adic Hodge       theory and their function theory, then explain a general formalism for studying these structures.