Notes: https://drive.google.com/file/d/1PWW_guqKGBvWsrnddjX18XdTKX2KbeLG/view?usp=sharing
Abstract: Prismatic F-gauges are the natural coefficient systems for prismatic cohomology, analogous to variations of Hodge structures in classical Hodge theory. This talk will describe a couple of equivalent perspectives on this notion, and then present evidence suggesting that prismatic F-gauges over Spf(Z_p) might provide a meaningful notion of crystallinity for representations of the absolute Galois group of Q_p with torsion coefficients. This is joint work in progress with Jacob Lurie, building on work of Drinfeld.
Spring 2022
Abstract: 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. 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 analogue of this conjecture for local Shimura varieties. This implies that, for p-adically uniformized global Shimura varieties, we obtain an action of the local plectic group on the level of complexes. The proof crucially uses Fargues–Scholze's results on the cohomology of moduli spaces of local shtukas.
Recording: https://bostonu.zoom.us/rec/share/rbHWfUAdZz5gFDYhPVL86mGJhJY8CfVU0KEuq-I_9ftuqdbrVMrLUro_Dk_f95M1.478wLaMwEfGNPKmD
Abstract: Investigating the p-adic integration map constructed by J.-M. Fontaine during the 90's, which is the main tool for proving the Hodge--Tate decomposition of the Tate module of an abelian variety over a p-adic field, we realized that the group of p-adic points of the above-named abelian variety, satisfying certain hypothesis, has a type of p-adic uniformization which was not remarked before. This is joint work with A. Iovita and A. Zaharescu.
Notes: https://drive.google.com/file/d/1jvgwjIB3u9NC8YniqO9rSP0f1QaccQqq/view?usp=sharing
Abstract: This second talk (based on joint work with Anschütz–Gleason–Richarz) concerns the Scholze–Weinstein conjecture on the representability of v-sheaf local models for geometric conjugacy classes of minuscule coweights. I'll start by reviewing previously known instances of local models in PEL cases by Rapoport–Zink, and also via power series Grassmannians by Pappas–Zhu. I'll briefly explain how to slightly refine the latter (joint with Fakhruddin–Haines–Richarz). Building on this, I'll explain the comparison of p-adic admissible loci in the Witt Grassmannian with those found in power series Grassmannians. Next, I'll prove the specialization principle for sufficiently nice kimberlites, which include v-sheaf local models (even for non-minuscule cocharacters). Finally, I'm going to explain how to compute the specialization mapping in families, deducing the Scholze–Weinstein conjecture.
Abstract: The first talk concerns the \'etale cohomology of the v-sheaf local models. After motivating the definition of v-sheaf local models we will determine their special fibers by calculating the nearby cycles of Satake sheaves.
Abstract: Let $p$ be a prime. I plan to explain how to read the $p$-adic Hodge structure of the $p$-adically completed cohomology of modular curves by studying the $p$-adic geometry of the modular curves at infinite level. One main tool is the relative Sen theory (also called $p$-adic Simpson correspondence) which provides a first-order differential equation and allows us to apply differential operators pulled back from the flag variety along the Hodge-Tate period map.
Lecture (1): Hodge-Tate structure Lecture (2): de Rham structure
If time permits, I will also discuss several applications.
Abstract: Let $p$ be a prime. I plan to explain how to read the $p$-adic Hodge structure of the $p$-adically completed cohomology of modular curves by studying the $p$-adic geometry of the modular curves at infinite level. One main tool is the relative Sen theory (also called $p$-adic Simpson correspondence) which provides a first-order differential equation and allows us to apply differential operators pulled back from the flag variety along the Hodge-Tate period map.
Lecture (1): Hodge-Tate structure Lecture (2): de Rham structure
If time permits, I will also discuss several applications.
Fall 2021
Abstract: We have developed local cohomology techniques to study the coherent cohomology of Shimura varieties. The local cohomology groups which appear are a generalization of overconvergent modular forms studied by Coleman and many others.
Tentative plan of the lectures : 1) Overview of the results and analogy with classical representation theory 2) Definition of the local cohomology, vanishing theorems and slope estimates. 3) Eigenvarieties and applications.
Abstract: We have developed local cohomology techniques to study the coherent cohomology of Shimura varieties. The local cohomology groups which appear are a generalization of overconvergent modular forms studied by Coleman and many others.
Tentative plan of the lectures : 1) Overview of the results and analogy with classical representation theory 2) Definition of the local cohomology, vanishing theorems and slope estimates. 3) Eigenvarieties and applications.
Abstract: We have developed local cohomology techniques to study the coherent cohomology of Shimura varieties. The local cohomology groups which appear are a generalization of overconvergent modular forms studied by Coleman and many others.
Tentative plan of the lectures : 1) Overview of the results and analogy with classical representation theory 2) Definition of the local cohomology, vanishing theorems and slope estimates. 3) Eigenvarieties and applications.
Spring 2021 (under construction)
Notes: https://math.bu.edu/people/jsweinst/rampage/Morra.pdf
Fall 2020 (under construction)