FGC-HRI-IPM Joint Number Theory Seminar

FGC, HRI and IPM are holding a joint biweekly zoom Number Theory seminar for Winter/Spring 2023.

When: Wednesdays
Where: Zoom
Meeting ID: 85613860958
Seminar mailing list: google group

Past Talks

Speaker: Olga Lukina
Title: Weyl groups in Cantor dynamics
Abstract: Arboreal representations of absolute Galois groups of number fields are given by profinite groups of automorphisms of regular rooted trees, with the geometry of the tree determined by a polynomial which defines such a representation. Thus arboreal representations give rise to dynamical systems on a Cantor set, and allow to apply the methods of topological dynamics to study problems in number theory. In this talk we consider the conjecture of Boston and Jones, which states that the images of Frobenius elements under arboreal representations have a certain cycle structure. To study this conjecture, we borrow from the Lie group theory the concepts of maximal tori and Weyl groups, and introduce maximal tori and Weyl groups in the profinite setting. We then use this new technique to give a partial answer to the conjecture by Boston and Jones in the case when an arboreal representations is defined by a post-critically finite quadratic polynomial over a number field. Based on a joint work with Maria Isabel Cortez.
Date: June 14, 2023
Video link
Passcode: 37*ta2if

Speaker: Carlo Pagano, Concordia University Montreal
Title: Abelian arboreal representations
Abstract: I will present joint work with Andrea Ferraguti which makes progress on a conjecture of Andrews and Petsche that classifies abelian dynamical Galois groups over number fields, in the unicritical case. I will explain how to reduce the conjecture to the post-critically finite case and the key tools to handle all unicritical PCF with periodic critical orbit over any number field and all PCF over quadratic number fields. Along the way I will present an earlier rigidity result of ours on the maximal closed subgroup of the automorphism group of a binary rooted tree, which offered us with the main input to translate the commutativity of the Galois image into diophantine equations. I will also overview progress on the tightly related problem of lower bounding arboreal degrees.
Date: May 31, 2023
Video link
Passcode: m5RS+P9W

Speaker: Cristiana Bertolin
Title: Periods of 1-motives and their polynomials relations
Abstract: The integration of differential forms furnishes an isomorphism between the De Rham and the Hodge realizations of a 1-motive M. The coefficients of the matrix representing this isomorphism are the so-called "periods" of M. In the semi-elliptic case (i.e. the underlying extension of the 1-motive is an extension of an elliptic curve by the multiplicative group), we compute explicitly these periods. If the 1-motive M is defined over an algebraically closed field, Grothendieck's conjecture asserts that the transcendence degree of the field generated by the periods is equal to the dimension of the motivic Galois group of M. If we denote by I the ideal generated by the polynomial relations between the periods, we have that "the numbers of periods of M minus the rank of the ideal I is equal to the dimension of the motivic Galois group of M", that is a decrease in the dimension of the motivic Galois group is equivalent to an increase of the rank of the ideal I. We list the geometrical phenomena which imply the decrease in the dimension of the motivic Galois group and in each case we compute the polynomials which generate the corresponding ideal I.
Date: May 17, 2023
Video link
Passcode: $df6.L*J

Speaker: Rahul Gupta, Harish-Chandra Research Institute
Title: Tame class field theory
Abstract: As a part of global class field theory, we construct a reciprocity map that describes the unramified (resp. tame) étale fundamental group as a pro-completion of a suitable idele class group (resp. tame idele class group) for smooth curves over finite fields. These results were extended to higher-dimensional smooth varieties over finite fields by Kato-Saito (unramified case, in 1986) and Schmidt-Spiess (tame case, in 2000). We begin the talk by recalling these results. The main focus of the talk is to work with smooth varieties over local fields. The class field theory over local fields is not as nice as that over finite fields. We discuss results in the unramified class field theory over local fields achieved in the period 1981–2015 by various mathematicians (Bloch, Saito, Jennsen, Forre, etc.). We then move to the main topic of the talk which is the tame class field theory over local fields and prove that the results in the tame case are similar to that in the case of unramified class field theory. This talk will be based on a joint work with A. Krishna and J. Rathore.
Date: May 3, 2023
Video link
Passcode: 1jdC6t&w

Speaker: Asgar Jamneshan, Koc University
Title: On inverse theorems and conjectures in ergodic theory and additive combinatorics
Abstract: I will provide a non-technical overview of some interactions between ergodic theory and additive combinatorics. The focus will be on inverse theorems and conjectures for the Gowers uniformity norms for finite abelian groups in additive combinatorics and their counterparts for the Host-Kra-Gowers uniformity seminorms for abelian measure-preserving systems in ergodic theory.
Date: April 19, 2023
Video link
Passcode: 5.?44?LT

Speaker: Alia Hamieh
Title: Moments of L-functions and Mean Values of Long Dirichlet Polynomials
Abstract: Establishing asymptotic formulae for moments of L-functions is a central theme in analytic number theory. This topic is related to various non-vanishing conjectures and and the generalized Lindelöf Hypothesis. A major breakthrough in analytic number theory occurred in 1998 when Keating and Snaith established a conjectural formula for moments of the Riemann zeta function using ideas from random matrix theory. The methods of Keating and Snaith led to similar conjectures for moments of many families of L-functions. These conjectures have become a driving force in this field which has witnessed substantial progress in the last two decades. In this talk, I will review the history of this subject and survey some recent results. I will also discuss recent joint work with Nathan Ng on the mean values of long Dirichlet polynomials which could be used to model moments of the zeta function.
Date: April 5, 2023
Video link
Passcode: %v?5DHG6

Speaker: Ilker Inam, Bilecik Üniversitesi
Title: Fast computation of half-integral weight modular forms
Abstract: Modular forms continue to attract attention for decades with many different application areas. To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. In this talk, we will show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations.
Date: March 15, 2023
Video link
Passcode: Z#%8V=cf