NTS ABSTRACTSpring2022

Jan 27

Daniel Li-Huerta

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. 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 includes (the generic fibers of) Lubin–Tate spaces, Drinfeld upper half spaces, and more generally Rapoport–Zink spaces. The proof crucially uses Scholze's theory of diamonds. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits. Recording for this talk is available upon request. Please email to zyang352@wisc.edu.

Feb 3

Weibo Fu

Sharp bounds for multiplicities of Bianchi modular forms

We prove a degree-one saving bound for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on SL_2 over any number field that is not totally real. In particular, we establish a sharp bound on the growth of cuspidal Bianchi modular forms. We transfer our problem into a question over the completed universal enveloping algebras by applying an algebraic microlocalisation of Ardakov and Wadsley to the completed homology. We prove finitely generated Iwasawa modules under the microlocalisation are generic, solving the representation theoretic question by estimating growth of Poincare–Birkhoff–Witt filtrations on such modules. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Feb 10

Marco D'Addezio

Parabolicity conjecture of F-isocrystals

I will talk about Crew's parabolicity conjecture for the algebraic monodromy groups of overconvergent F-isocrystals. Besides the proof, I will explore the main consequences of this conjecture. For example, I will explain how to deduce from the conjecture that over finitely generated fields of positive characteristic p the Galois action on the étale p-adic Tate module of an abelian variety is semi-simple. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Feb 17

Haoyang Guo

Hodge-Tate Decomposition

In complex geometry, one of the most fundamental results is the Hodge decomposition, which builds a bridge between the underlying topological information and the algebraic/differential geometric information of a given smooth complex variety. The analogous result in p-adic geometry, conjectured by Tate and proved by Faltings and many others, is called the Hodge-Tate decomposition. It states that as a Galois representation, p-adic etale cohomology of a p-adic smooth variety decomposes into a direct sum of Hodge cohomology. In particular, this allows us to encode the Galois representational structure by algebraic geometry. In this talk, we will discuss this decomposition, and consider its generalization to non-smooth varieties. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Feb 24

Pol van Hoften

Hodge-Tate Decomposition

A classical theorem of Chai says that the prime-to-p Hecke orbit of an ordinary point in the moduli space of principally polarized abelian varieties over a finite field is Zariski dense in the whole moduli space. This talk is about an extension of this result to Shimura varieties of Hodge type. The proof makes use of the global Serre-Tate coordinates of Chai as well as recent results of D'Addezio about the $p$-adic monodromy of isocrystals. If time permits, I will discuss how our strategy might be used to tackle more cases of the Hecke-orbit conjecture. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.