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

On the ordinary Hecke-orbit conjecture

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.

Mar 03

Nina Zubrilina

Convergence to Plancherel measure of Hecke eigenvalues

Joint work with Peter Sarnak. We give rates, uniform in the degrees of test polynomials, of convergence of Hecke eigenvalues to the p-adic Plancherel measure. We apply this to the question of eigenvalue tuple multiplicity and to a question of Serre concerning the factorization of the Jacobian of the modular curve X_0(N). Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Mar 10

Will Chen

Markoff triples and connectedness of Hurwitz spaces

Let G be a finite group, and let H_{g,n,G} denote the Hurwitz space of G-covers of genus g curves with n branch points. It is a classical problem to classify the connected components of H_{g,n,G} using geometric invariants of covers. If one fixes G and allows g or n to be large, results of Conway-Parker and Dunfield-Thurston give satisfying descriptions of the connected components. However, if one fixes (g,n) and allows G to vary over an infinite family of highly nonabelian groups, then much less in known. In this talk we will show that the substack of H_{1,1,SL(2,p)} classifying covers with ramification indices 2p is connected for large p. The proof combines estimates of Bourgain, Gamburd, and Sarnak with an additional rigidity coming from algebraic geometry. This yields a strong approximation property for the Markoff equation, and with at most finitely many exceptions, resolves a question of Frobenius from 1913 on congruences satisfied by Markoff numbers. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Mar 17

Will Sawin

Equidistribution of CM points, primes in progressions, Vanishing cycles, and the characteristic cycle

This talk covers two different problems in number theory. One, suggested by Michel and Venkatesh, is about the distribution of CM points on the product of two modular curves and is closely related to the Andre-Oort conjecture. The other is about the number of primes in an arithmetic progression. Both problems have analogues in the function field context, and, though they seem totally unrelated, the same geometric strategy can be used to attack both. I will explain this connection. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Mar 24

Itay Glazer

On singularity properties of word maps and applications to probabilistic Waring type problems.

Given two morphisms f and g from algebraic varieties X and Y to an algebraic group G, we define their convolution to be the morphism f∗g from X×Y to G by f∗g(x,y):=f(x)g(y). Similarly to the smoothing effect of the convolution operation in analysis, this operation yields morphisms with improved singularity properties. Given a word w in a free group F_r on a set of r elements, and an algebraic group G, one can associate a word map w:G^r-->G (e.g. the commutator map (x,y)--->[x,y]). We apply the above philosophy and show that word maps on semisimple Lie groups and Lie algebras have nice singularity properties after sufficiently many self-convolutions, with bounds depending only on the complexity of the word. The singularity properties we consider are intimately connected to the point count of schemes over finite rings of the form Z/p^kZ. We utilize this connection to provide applications in group theory, namely to the study of random walks on compact p-adic groups induced by these word maps. The talk is (mostly) based on a joint work with Yotam Hendel https://arxiv.org/abs/1912.12556. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.