Time and Room: Monday 1:30 pm -3:00 pm BRNG 1206
Schedule:
Aug 24th: Speaker: Tong Liu : Overview BGK paper
Aug 31th: Speaker: Michael Wills: Review of crystalline and semisimple mod p representations of G_Q_p
Title: Introduction to crystalline and mod-p representations
Abstract: Arbitrary representations over Z_ell of the absolute Galois group of Q_p can be difficult to characterize when ell = p, in part because their reduction modulo p does not preserve the action of wild inertia. For an interesting subcategory of these, the crystalline representations, the action of inertia may be described via a filtration on a Qp-vector space. A recent preprint of Bhatt--Gee--Kisin describes a connection between a crystalline representation's associated filtration and the representation's reduction modulo p. This talk is the second in a learning seminar aimed at understanding this preprint, and will introduce the basics of both crystalline and mod-p representations of Q_p.
Sept 14th: Speaker: Ruipeng Zou: Affine Weyl groups
Title: Up-arrow ordering and affine Grassmannians
Abstract: Let $G$ be a split connected reductive group. We will go over the up arrow order on coweights and alcoves in the affine Weyl group. We explain its relationship with the Bruhat order, including Wang's theorem identifying the upper-arrow order on dominant alcoves with Bruhat order, and the resulting description of the upper-arrow order on dominant coweights. We will also review the affine Grassmannian and study the $\mu_p$-fixed locus in the affine Grassmannian.
Sept 21st: Speaker: Ruipeng Zhou : Affine Grassmannin
Title: \mu_p-fixed locus and orbit closure of affine Grassmannians
Abstract: Let G be a split connected reductive group. In this talk, we will review the affine Grassmannian of G, its Cartan and Iwahori decompositions, and the associated Schubert stratifications. We will then study the \mu_p-fixed locus using affine flag varieties and parahoric subgroups. Finally, we will discuss the orbit-closure results using the upper-arrow order.
Sept 28th: Speaker: Varun Balasubramanian: Etale \varphi-modules
Title: Etale phi-modules
Abstract: Consider an infinitely ramified extension K_infinity obtained by adjoining a compatible system of p^n^th powers of p to Q_p. Our goal is to understand the Z_p representations of the absolute Galois group of G_K_infinity. A result of Fontaine and Wintenburger showed that G_{K_infinity} is canonically isomorphic to G_E=Gal(E^sep/E), where E=F_p((u)). So in this talk, we'll study finite dimensional F_p and Z_p representations of G_E by relating them to (semi-)linear algebraic data, which are known as etale phi-modules. We'll work a set of 1-dimensional and 2-dimensional representations to illustrate the theory which'll be built in this talk.
Oct. 5th: Speaker: Boqiang Zhang: Introduction of Kisin modules.
Title: Introduction to Breuil–Kisin modules
Abstract: The goal of this talk is to introduce Breuil–Kisin modules and their relation to crystalline Galois lattices and their reductions modulo $p$. We will discuss Kisin's fully faithful functor from crystalline $G_K$-lattices to Breuil–Kisin modules. Using the theory of étale $\varphi$-modules from the previous talk, we will then explain how the associated module recovers the original lattice after restriction to $G_{K_\infty}$. Finally, we will explain the compatibility of this comparison with reduction modulo $p$ and briefly indicate its role in the work of Bhatt–Gee–Kisin.
Oct. 19th: μ_p structure. Speaker: George Nicolas Diaz-Wahl
Title: mu_p equivariance for Breuil-Kisin modules.
Abstract: We define a \mu_p-action on the Breuil-Kisin prism and state a theorem regarding \mu_p-equivariance for alterations of Breuil-Kisin modules. This result gives a precise relationship between the Hodge filtration on a Breuil-Kisin module and its mod p reduction. This is technical key to controlling the Hodge-Tate weights of crystalline lifts of a mod p-Galois representation.
Oct. 26th: μ_p structure (II). Speaker: George Nicolas Diaz-Wahl
Title: mu_p equivariance for Breuil-Kisin modules (II).
Abstract: Will complete the story from the last lecture.
Nov 2nd: Review of prismatrization of O_K. Speaker: Michael Wills
Title: Prismatization.
Abstract: Previously, we saw that the Breuil-Kisin module of a crystalline representation carries information about the Hodge-Tate weights and inertial weights of that representation. In this talk, we will define the prismatization of Z_p (a geometric object parametrizing generalized prism structures on p-nilpotent Z_p algebras) and explain how this gives a new perspective on the Breuil-Kisin module. This continues the ongoing learning seminar on a paper of Bhatt-Gee-Kisin.
Nov. 9th: The canonical modification via the prismatization. Speaker: Daniel Le
Title: Canonical modifications of crystalline Breuil—Kisin modules
Abstract: We explain the construction of a modification of a crystalline Breuil—Kisin module using prismatic F-crystals (as a black box) and a comparison of the Hodge—Tate and Breuil--Kisin Cartier--Witt divisors.
Nov. 16th: Theorem 5.5.6. The semi-simplification of reduction of crystalline representation. Speaker: Tong Liu
Title: The tame inertia weights of the reduction of crystalline representations
Abstract: We will collect the preparations from this semester to prove the main theorem of Bhatt-Gee-Kisin's paper: How to determine the tame inertia weights of the reduction of a crystalline representation by its Hodge-Tate weights.
Nov.23rd. No talk or free discussion.
Nov 30th. An optimistic conjecture on Serre's weights. Speaker: Daniel Le
References:
Bhatt Gee and Kisin: Reduction modulo p of crystalline Galois representations via μ_p-equivariance.