NTS ABSTRACTSpring2020

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Jan 23

Rahul Krishna

A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups

The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.

Jan 30

Eric Stubley

Class Groups, Congruences, and Cup Products

The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.

Feb 6

Brian Smithling

On Shimura varieties for unitary groups

Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.

Feb 20

Shai Evra

Ramanujan Conjectures and Density Theorems

The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program.

Feb 27

Mathilde Gerbelli-Gauthier

Cohomology of Arithmetic Groups and Endoscopy

How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.