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.

Feb 2

Daniel Li-Huerta

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 Poincar\'e–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.