Seminars Sorted by Series

Joint IAS/Princeton University Number Theory Seminar

Jan
23
2020

Joint IAS/Princeton University Number Theory Seminar

Motivic Euler products in motivic statistics
Margaret Bilu
4:30pm|Princeton University, Fine Hall 214

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties...

Jan
30
2020

Joint IAS/Princeton University Number Theory Seminar

Eisenstein series and the cubic moment for PGL(2)
Paul Nelson
4:30pm|Simonyi Hall 101

We will discuss how to study the cubic moment of any family of automorphic L-functions on PGL(2) using regularized diagonal periods of Eisenstein series, following a strategy suggested by Michel--Venkatesh. Applications include generalizations to...

Feb
06
2020

Joint IAS/Princeton University Number Theory Seminar

Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture
Daniel Kriz
4:30pm|Princeton University, Fine Hall 214

In this talk, I formulate and prove a new Rubin-type Iwasawa main conjecture for imaginary quadratic fields in which p is inert or ramified, as well as a Perrin-Riou type Heegner point main conjecture for certain supersingular CM elliptic curves...

Feb
13
2020

Joint IAS/Princeton University Number Theory Seminar

Moduli spaces of shtukas over function fields
4:30pm|Simonyi Hall 101

We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some...

Feb
20
2020

Joint IAS/Princeton University Number Theory Seminar

Isolation of L^2 spectrum and application to Gan-Gross-Prasad conjecture
Yifeng Liu
4:30pm|Princeton University, Fine Hall 214

In this talk, we will introduce a new technique of isolating the cuspidal part, or more general cuspidal components, from the $L^2$ spectrum on the spectral side of the trace formula. As an application, we prove the global Gan-Gross-Prasad and...

Feb
27
2020

Joint IAS/Princeton University Number Theory Seminar

A p-adic monodromy theorem for de Rham local systems
4:30pm|Simonyi 101

Every smooth proper algebraic variety over a p-adic field is expected to have semistable model after passing to a finite extension. This conjecture is open in general, but its analogue for Galois representations, the p-adic monodromy theorem, is...

Mar
05
2020

Joint IAS/Princeton University Number Theory Seminar

Euler system, Eisenstein congruences and the p-adic Langlands correspondence
Eric Urban
4:30pm|Princeton University, Fine Hall 214

I will discuss how the use of the p-adic Langlands correspondence for GL_2(Q_p) allows to study Eisenstein congruences of various weight, level and slopes in order to construct Euler systems. I will discuss the GL(2)-case with some details and give...

Mar
26
2020

Joint IAS/Princeton University Number Theory Seminar

Wiles defect for Hecke algebras that are not complete intersections
4:30pm|https://theias.zoom.us/j/280491607

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. In addition to proving modularity theorems, this numerical criterion also implies a connection between the...

Apr
02
2020

Joint IAS/Princeton University Number Theory Seminar

Density conjecture for horizontal families of lattices in SL(2)
4:30pm|https://theias.zoom.us/j/959183254

Let G be a real semi-simple Lie group with an irreducible unitary representation \pi. The non-temperedness of \pi is measured by the parameter p(\pi) which is defined as the infimum of p\geq 2 such that \pi has matrix coefficients in L^p(G). Sarnak...

Apr
09
2020

Joint IAS/Princeton University Number Theory Seminar

On the Kudla-Rapoport conjecture
4:30pm|https://theias.zoom.us/j/959183254

The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. It is a key local...

Apr
16
2020

Joint IAS/Princeton University Number Theory Seminar

Local-global compatibility in the crystalline case
3:00pm|https://theias.zoom.us/j/959183254

Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with p-torsion coefficients. These Galois representations are expected to satisfy local-global...

Apr
23
2020

Joint IAS/Princeton University Number Theory Seminar

Symmetric power functoriality for holomorphic modular forms
Jack Thorne
9:00am|https://theias.zoom.us/j/959183254

Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2), thought of as the L-group...

Apr
30
2020

Joint IAS/Princeton University Number Theory Seminar

Eulerianity of Fourier coefficients of automorphic forms
Henrik Gustafsson
4:30pm|https://theias.zoom.us/j/959183254

The factorization of Fourier coefficients of automorphic forms plays an important role in a wide range of topics, from the study of L-functions to the interpretation of scattering amplitudes in string theory. In this talk I will present a transfer...

May
07
2020

Joint IAS/Princeton University Number Theory Seminar

On triple product L functions
Jayce Robert Getz
4:30pm|https://theias.zoom.us/j/959183254

Establishing the conjectured analytic properties of triple product L-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on GL_3; in...

May
14
2020

Joint IAS/Princeton University Number Theory Seminar

A geometric view on Iwasawa theory
Mladen Dimitrov
2:30pm|https://theias.zoom.us/j/959183254

We will investigate the geometry of the p-adic eigencurve at classical points where the Galois representation is locally trivial at p, and will give applications to Iwasawa and Hida theories.

May
21
2020

Joint IAS/Princeton University Number Theory Seminar

Iwasawa theory and Bloch-Kato conjecture for unitary groups
9:00am|https://theias.zoom.us/j/959183254

We describe a new method to study Eisenstein family and Iwasawa theory on unitary groups over totally real fields of general signatures. As a consequence we prove that if the central L-value of a cuspidal eigenform on the unitary group twisted by a...

May
28
2020

Joint IAS/Princeton University Number Theory Seminar

Joint equidistribution of adelic torus orbits and families of twisted L-functions
10:00am|https://theias.zoom.us/j/959183254

The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful...

Jun
04
2020

Joint IAS/Princeton University Number Theory Seminar

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions
Florian Richter
3:00pm|https://theias.zoom.us/j/959183254

 

One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new...

Jun
11
2020

Joint IAS/Princeton University Number Theory Seminar

New constraints on the Galois configurations of algebraic integers in the complex plane
3:00pm|https://theias.zoom.us/j/959183254

Fekete (1923) discovered the notion of transfinite diameter while studying the possible configurations of Galois orbits of algebraic integers in the complex plane. Based purely on the fact that the discriminants of monic integer irreducible...

Jun
18
2020

Joint IAS/Princeton University Number Theory Seminar

Independence of ℓ for Frobenius conjugacy classes attached to abelian varieties
3:00pm|https://theias.zoom.us/j/959183254

Let $A$ be an abelian variety over a number field $E\subset \mathbb{C}$ and let $v$ be a place of good reduction lying over a prime $p$. For a prime $\ell\neq p$, a result of Deligne implies that upon replacing $E$ by a finite extension, the Galois...

Sep
10
2020

Joint IAS/Princeton University Number Theory Seminar

An asymptotic version of the prime power conjecture for perfect difference sets
4:30pm|Simonyi 101 and Remote Access

A subset $D$ of a finite cyclic group $Z/mZ$ is called a "perfect difference set" if every nonzero element of $Z/mZ$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that...

Sep
17
2020

Joint IAS/Princeton University Number Theory Seminar

Equivariant localization, parity sheaves, and cyclic base change
2:00pm|Remote Access

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. I will explain some recently established properties of these correspondences...

Sep
24
2020

Joint IAS/Princeton University Number Theory Seminar

A non-archimedean definable Chow theorem
4:30pm|Remote Access

In recent years, o-minimality has found some striking applications to diophantine geometry. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the...

Oct
08
2020

Joint IAS/Princeton University Number Theory Seminar

Representations of p-adic groups and applications
2:00pm|Remote Access

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the...

Oct
15
2020

Joint IAS/Princeton University Number Theory Seminar

Heights and dynamics over arbitrary fields
Carney Alexander
4:30pm|Remote Access

Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and...

Oct
22
2020

Joint IAS/Princeton University Number Theory Seminar

On the locally analytic vectors of the completed cohomology of modular curves
Lue Pan
4:30pm|Remote Access

A classical result identifies holomorphic modular forms with highest weight vectors of certain representations of $SL_2(\mathbb{R})$. We study locally analytic vectors of the (p-adically) completed cohomology of modular curves and prove a p-adic...

Oct
29
2020

Joint IAS/Princeton University Number Theory Seminar

An explicit supercuspidal local Langlands correspondence
4:30pm|Remote Access

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any $p$-adic group $G$ that splits over a tame extension, provided $p$ does not divide the order of the Weyl group. This construction matches...

Nov
05
2020

Joint IAS/Princeton University Number Theory Seminar

Strong approximation for the Markoff equation via nonabelian level structures on elliptic curves
William Chen
4:30pm|Remote Access

Following Bourgain, Gamburd, and Sarnak, we say that the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$ satisfies strong approximation at a prime $p$ if its integral points surject onto its $F_p$ points. In 2016, Bourgain, Gamburd, and Sarnak were...

Nov
12
2020

Joint IAS/Princeton University Number Theory Seminar

Effective height bounds for odd-degree totally real points on some curves
Levent Alpoge
4:30pm|Remote Access

Let \o be an order in a totally real field, say F. Let K be an odd-degree totally real field. Let S be a finite set of places of K. We study S-integral K-points on integral models H_\o of Hilbert modular varieties because not only do said varieties...

Nov
19
2020

Joint IAS/Princeton University Number Theory Seminar

Ramanujan Conjecture and the Density Hypothesis
4:30pm|Remote Access

The Generalized Ramanujan Conjecture (GRC) for GL(n) is a central open problem in modern number theory. Its resolution is known to yield several important applications. For instance, the Ramanujan-Petersson conjecture for GL(2), proven by Deligne...

Dec
03
2020

Joint IAS/Princeton University Number Theory Seminar

A unitary analogy of Friedberg-Jacquet and Guo-Jacquet periods and central values of standard L functions on GL(2n)
4:30pm|Remote Access

Let $G$ be a reductive group over a number field $F$ and $H$ a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on $G$ over $H(F)\backslash H(A_F)$. They are often related to special values of certain L functions. One...

Dec
10
2020

Joint IAS/Princeton University Number Theory Seminar

On the Liouville function at polynomial arguments
Joni Teräväinen
4:30pm|Remote Access

Let $\lambda$ be the Liouville function and $P(x)$ any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence $\lambda(P(n))$ changes sign infinitely often. We present a solution to this...

Jan
21
2021

Joint IAS/Princeton University Number Theory Seminar

Ax-Lindemann-Weierstrass Theorem (ALW) for Fuchsian automorphic functions
Joel Nagloo
4:30pm|Remote Access

Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points...

Feb
11
2021

Joint IAS/Princeton University Number Theory Seminar

Cohomology of Arithmetic Groups and Endoscopy
4:30pm|Remote Access

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...

Feb
18
2021

Joint IAS/Princeton University Number Theory Seminar

Exceptional jumps of Picard rank of K3 surfaces over number fields
4:30pm|Remote Access

Given a K3 surface $X$ over a number field $K$, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite, assuming that $X$ has everywhere potentially good reduction. This result is formulated in the general framework...

Feb
25
2021

Joint IAS/Princeton University Number Theory Seminar

Selmer groups and a Cassels-Tate pairing for finite Galois modules
Alexander Smith
4:30pm|Remote Access

I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the...

Mar
04
2021

Joint IAS/Princeton University Number Theory Seminar

Monoidal Structures on GL(2)-Modules and Abstractly Automorphic Representations
Gal Dor
4:30pm|Remote Access

Consider the function field F of a smooth curve over $F_q$, with $q > 2$.

L-functions of automorphic representations of $GL(2)$ over $F$ are important objects for studying the arithmetic properties of the field $F$. Unfortunately, they can be...

Mar
18
2021

Joint IAS/Princeton University Number Theory Seminar

The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties
Will Sawin
4:30pm|Remote Access

Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension $n$, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In...

Mar
25
2021

Joint IAS/Princeton University Number Theory Seminar

The local Gan-Gross-Prasad conjecture for real unitary groups
4:30pm|Remote Access

A classical branching theorem of Weyl describes how an irreducible representation of compact $U(n+1)$ decomposes when restricted to $U(n)$. The local Gan-Gross-Prasad conjecture provides a conjectural extension to the setting of representations of...

Apr
01
2021

Joint IAS/Princeton University Number Theory Seminar

Eisenstein series, p-adic deformations, Galois representations, and the group G_2
Sam Mundy
4:30pm|Remote Access

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs nontrivial elements in the Selmer groups of...

Apr
08
2021

Joint IAS/Princeton University Number Theory Seminar

Low moments of character sums
Adam Harper
4:30pm|Remote Access

Sums of Dirichlet characters $\sum_{n \leq x} \chi(n)$ (where $\chi$ is a character modulo some prime $r$, say) are one of the best studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the...

Apr
15
2021

Joint IAS/Princeton University Number Theory Seminar

Beilinson-Bloch conjecture for unitary Shimura varieties
4:30pm|Remote Access

For certain automorphic representations $\pi$ on unitary groups, we show that if $L(s, \pi)$ vanishes to order one at the center $s=1/2$, then the associated $\pi$-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of...

Apr
22
2021

Joint IAS/Princeton University Number Theory Seminar

Kolyvagin's conjecture and higher congruences of modular forms
Naomi Sweeting
4:30pm|Remote Access

Given an elliptic curve $E$, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of $E$. Under the conjecture that not all of these classes vanish, he gave a description...