Zoom link:
https://ubc.zoom.us/j/9396078403?pwd=uD6PhmyITsV6zRioCOXydK1YTgDB0p.1
password: 004133

 Clifton Cunningham
 
Title: Problems arising (and some solutions) from Vogan's conception
of the local Langlands correspondence
Abstract: David Vogan's 1993 paper on the local Langlands
correspondence showed that there are advantages to simultaneously
considering representations of a p-adic group together with its pure
inner forms. Combined with other insights, this opened the door to
introducing ABV-packets for p-adic groups, which provide a vast
generalization of A-packets, and also suggested questions about the
structure of the category of perverse sheaves on a variety of certain
Langlands parameters. In this talk we survey recent developments on
these matters, including: ABV-packets for non-Arthur type Langlands
parameters of GL(N); ABV-packets for unipotent representations of G(2);
and comparing the category of perverse sheaves, above, with the Koszul
dual of the corresponding generalized Steinberg representations. We will
also discuss several problems not yet resolved that are on the horizon.
Location: ESB 4133 (only)

Didier Lesesvre
Note the time change: Friday 22nd, 12:30pm

Title: Relation between low-lying zeros and central values
Abstract: In practice, L-functions appear as generating functions
encapsulating information about various objects, such as Galois
representations, elliptic curves, arithmetic functions, modular forms,
Maass forms, etc. Studying L-functions is therefore of utmost importance
in number theory at large. Two of their attached data carry critical
information: their zeros, which govern the distributional behavior of
underlying objects; and their central values, which are related to
invariants such as the class number of a field extension. We discuss a
connection between low-lying zeros and central values of L-functions, in
particular showing that results about the distribution of low-lying
zeros (towards the density conjecture of Katz-Sarnak) implies results
about the distribution of the central values (towards the normal
distribution conjecture of Keating-Snaith). Even though we discuss this
principle in general, we instanciate it in the case of modular forms in
the level aspect to give a statement and explain the arguments of the proof.
Location: Online

Title: The Ostrowski Quotient for a finite extension of number fields

Abstract: For a number field $K$, the P\'olya group of $K$, denoted by $Po(K)$,
is the subgroup of the ideal class group of $K$ generated by the classes of the
products of maximal ideals of $K$ with the same norm. In this talk, after
reviewing some results concerning $Po(K)$, I will generalize this notion to the
relative P\'olya group $Po(K/F)$, for $K/F$ a finite extension of number
fields. Accordingly, I will generalize some results in the literature about
P\'olya groups to the relative case. Then, due to some essential observations,
I will explain why we need to modify the notion of the relative P\'olya group
to the Ostrowski quotient $Ost(K/F)$ to get a more 'accurate' generalization of
$Po(K)$. The talk is based on a joint work with Ali Rajaei (Tarbiat Modares
University) and Ehsan Shahoseini (Institute For Research In Fundamental
Sciences).

Emily Quesada-Herrera

Title:
On the vertical distribution of the zeros of the Riemann zeta-function

 Abstract:
In 1973, assuming the Riemann hypothesis (RH), Montgomery studied the vertical
distribution of zeta zeros, and conjectured that they behave like the
eigenvalues of some random matrices. We will discuss some models for zeta zeros
– starting from the random matrix model but going beyond it – and related
questions, conjectures and results on statistical information on the zeros. In
particular, assuming RH and a conjecture of Chan for how often gaps between
zeros can be close to a fixed non-zero value, we will discuss our proof of a
conjecture of Berry (1988) for the number variance of zeta zeros, in a regime
where random matrix models alone do not accurately predict the actual behavior
(based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).

Title: Decomposition of cohomology classes in finite field extensions

Abstract: Rost and Voevodsky proved the Bloch-Kato conjecture relating
Milnor k-theory and Galois cohomology. It implies that if a field F
contains a primitive pth root of unity, then the Galois cohomology ring of
F with coefficients in the trivial F-module with p elements is generated by
elements of degree one. In this talk, I will discuss a systematic approach
to studying this phenomenon in finite field extensions via decomposition
fields. This is joint work with Sunil Chebolu, Jan Minac, Cihan Okay, and
Andrew Schultz.

Manish Patnaik 

Title: Borel—Serre type constructions for Loop Groups
     
Abstract: (Joint work with Punya Satpathy) For a reductive group G, Borel
and Serre introduced a compactification of a large class of arithmetic
quotients of the symmetric space attached to G. After reviewing some
aspects of their construction, we explain how to generalize it to the case
when G is replaced by an infinite-dimensional analogue LG, the loop group
of G. Along the way, we describe  a partition of an arithmetic quotient of
LG, inspired by the work of P.-H. Chaudouard for GL_n and related to
earlier constructions of Harder-Narasimhan and Behrend.

Nicolas Dupré

Title: Homotopy classes of simple pro-p Iwahori-Hecke modules

Abstract: Let G be a p-adic reductive group and k a field of
characteristic p. The category Rep(G) of smooth k-linear representations
is at the heart of the mod-p Langlands program. If we let I be a pro-p
Iwahori subgroup of G, there is an associated convolution algebra
H=k[I\G/I], called the pro-p Iwahori-Hecke algebra of G, that is
well-understood. Taking I-invariants then defines a functor U from
Rep(G) to the category Mod(H) of H-modules, which is expected to provide
a strong relationship between the two categories. However, aside from
small cases, the functor U fails to be exact and its behaviour remains
quite mysterious. In earlier joint work with J. Kohlhaase, we used the
language of model categories to study this situation. In that approach,
one instead studies the interplay between certain associated homotopy
categories of representations and H-modules. In this talk, I will give
an overview of this new approach and describe when distinct simple
(supersingular) H-modules can become homotopy equivalent to one another.