NTS ABSTRACTSpring2022

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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.

Mar 31

Amina Abdurrahman

A global cohomological formula for the square class of the central value of a symplectic L-function

In the 70s Deligne gave a topological formula for the local epsilon factors attached to an orthogonal representation. We consider the case of a symplectic representation and present a conjecture giving a topological formula for a finer invariant, the square class of its central value. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds, and give a sketch of the proofs. This is joint work with Akshay Venkatesh. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Apr 07

Alexander Petrov

On arithmetic characterization of local systems of geometric origin

I will talk about the problem of classifying local systems of geometric origin on algebraic varieties (that is, local systems arising as a subquotient of the relative cohomology of a family of varieties). Conjecture: For a smooth algebraic variety S over a finitely generated field F, a semi-simple Q_l-local system on S_{\bar{F}} is of geometric origin if and only if it extends to a local system on S_{F'} for a finite extension F'\supset F The 'only if' direction follows from the formalism of etale cohomology by a spreading out argument. My main goal will be to provide motivation for this conjecture arising from the Fontaine-Mazur conjecture and a p-adic Riemann-Hilbert correspondence, and discuss the implications of these techniques on the structure of the Galois action on the pro-algebraic completion of the etale fundamental group. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Apr 14

Elad Zelingher

On regularization of integrals of matrix coefficients associated to spherical Bessel models

The Gan-Gross-Prasad conjecture relates a special value of an L-function of two cuspidal automorphic representations to the non-vanishing of a certain period. The Ichino-Ikeda conjecture is a refinement of this conjecture. It roughly states that the absolute value of the square of the period in question can be expressed as a product of the special value of the L-function and a product of normalized local periods. However, in order to formulate this conjecture, one needs to assume that the representations in question are tempered everywhere, or else the convergence of the local periods is not guaranteed. The generalized Ramanujan conjecture speculates that the representations in question (cuspidal automorphic representations lying in generic packets) are already tempered everywhere. However, the generalized Ramanujan conjecture is far from being known. In this talk, I will explain how to drop the assumption that the representations are tempered almost everywhere. I will explain how to extend the definition of the normalized local periods for places where the local components are given by principal series representations. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Apr 21

Zahi Hazan

An Identity Relating Eisenstein Series on General Linear Groups

Eisenstein series are key objects in the theory of automorphic forms. They play an important role in the study of automorphic L-functions, and they figure out in the spectral decomposition of the L^2-space of automorphic forms. In recent years, new constructions of global integrals generating identities relating Eisenstein series were discovered. In 2018 Ginzburg and Soudry introduced two general identities relating Eisenstein series on split classical groups (generalizing Mœglin 1997, Ginzburg-Piatetski-Shapiro-Rallis 1997, and Cai-Friedberg-Ginzburg-Kaplan 2016), as well as double covers of symplectic groups (generalizing Ikeda 1994, and Ginzburg-Rallis-Soudry 2011). We consider the Kronecker product map of two general linear groups, $\mathrm{GL}_{m}(𝔸)$ and $\mathrm{GL}_{n}(𝔸)$, to $\mathrm{GL}_{mn}(𝔸)$. Now, similarly to Ginzburg and Soudry's construction, we use a degenerate Eisenstein series of $\mathrm{GL}_{mn}(𝔸)$ as a kernel function on $\mathrm{GL}_{m}(𝔸) \otimes \mathrm{GL}_{n}(𝔸)$. Integrating it against a cusp form on $\mathrm{GL}{n}(𝔸)$, we obtain a 'semi-degenerate' Eisenstein series on $\mathrm{GL}_{m}(𝔸)$. Locally, we find an interesting relation to the local Godement-Jacquet integral. This construction demonstrates the rise of interesting L-functions from integrals of doubling type, as suggested by the philosophy of Ginzburg and Soudry. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

Apr 28

Lea Beneish

Degree $d$ points on curves

Given a plane curve $C$ defined over $\mathbb{Q}$, when the genus of the curve is greater than one, Faltings' theorem tells us that the set of rational points on the curve is finite. It is then natural to consider higher degree points, that is, points on this curve defined over fields of degree $d$ over $\mathbb{Q}$. We ask for which natural numbers $d$ are there points on the curve in a field of degree $d$. There is a lot of structure in the set of values $d$, some of which we will explain in this talk. This talk is based on joint work with Andrew Granville. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.

May 5

Gyujin Oh

A cohomological approach to harmonic Maass forms

We interpret a harmonic Maass form as a variant of a local cohomology class of the modular curve. This is not only amenable to algebraic interpretation, but also nicely generalized to other Shimura varieties, avoiding the barrier of Koecher's principle. We exhibit how this goes in the case of Hilbert modular varieties and connect it with the existing literature. Zoom ID: 947 2112 8091 Password: The smallest prime > 200 (resp. >300) is the first (resp. last) 3 digits.