**FGC-HRI-IPM** Joint Number Theory Seminar

**When:**Tuesdays

**Where:**Zoom

**Seminar mailing list:**google group

## Talks

** Speaker: **Abbas Maarefparvar (IPM, Iran)

** Title: ** * On BRZ exact sequence for finite Galois extensions of number fields
*

** Abstract:** In this talk, I will shortly explain how to use some cohomological results of Brumer-Rosen and
Zantema to obtain a four-term exact sequence, called ``BRZ’’ standing for these authors, which reveals some
information about strongly ambiguous ideal classes (coinciding with relative Polya group) of a finite Galois
extension of number fields. As an application of the BRZ, I will reprove some well known results in the
literature. Then, as a minor modification on relative Polya group for a finite extension of number fields,
I will introduce the notion of ``relative Ostrowski quotient'' and give some new approaches of the BRZ
exact sequence. The main part of my talk is concerning a joint work with Ali Rajaei and Ehsan Shahoseini.

** Date:** June 14, 2022

** Speaker: **Somnath Jha (IIT Kanpur, India)

** Title: ** * Fine Selmer group of elliptic curves over global fields*

** Abstract:** The (p-infinity) fine Selmer group (also called the 0-Selmer group) of an elliptic curve
is a subgroup of the usual p-infinity Selmer group of an elliptic curve and is related to the first and
the second Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group
over the cyclotomic Z_p extension of a number field K is intricately related to Iwasawa's \mu-invariant
vanishing conjecture on the growth of p-part of the ideal class group of K in the cyclotomic tower.
In this talk, we will discuss the structure and properties of the fine Selmer group over certain p-adic
Lie extensions of global fields. This talk is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.

** Date:** May 31, 2022

Video link

** Speaker: ** Hamza Yesilyurt (Bilkent University, Ankara, Turkey)

** Title:** *A Modular Equation of Degree 61 *

** Abstract:** A modular equation of degree n is an equation that relates classical
theta functions with arguments q and q^n. The theory of modular equations started with
the works of Landen, Jacobi and Legendre. The theory gained popularity again with enormous
contributions made by Ramanujan. In this talk we will give a brief introduction to the
theory of modular equations and then obtain a new modular equation of degree 61 by
using a generalization of a theta function identity due to David M. Bressoud. This is a joint work with Ahmet Güloğlu.

**Date:** May 17, 2022

Video link

** Speaker: ** Chirantan Chowdhury (University of Duisburg-Essen)

** Title:** * Motivic Homotopy Theory of Algebraic Stacks *

** Abstract:** In this talk, we extend the stable motivic homotopy theory from schemes to a large class of algebraic
stacks using the language of ∞-categories. This includes quasi-separated algebraic spaces, local
quotient stacks (for example moduli of vector bundles). We also extend the six functor formalism
and prove statements like localization and purity.
We start the talk by motivating the need of such specific class of algebraic stacks which shall lead up
to the construction of the stable homotopy functor and the four functors f∗
, f∗, − ⊗ − and
Hom(−, −). We then briefly explain the motivation of ”Enhanced operation map” due to Liu and
Zheng which helps us to construct the exceptional functors. The talk ends by stating the other
properties of these six functors like localization and purity.

**Date:** April 19, 2022

Video link

**Speaker:**Farzad Aryan (Göttingen University)

**Title:**

*On the Riemann Zeta Function*

**Abstract :**I will discuss the Riemann zeta function and the significance of its zeros to prime numbers. Also, I will look at the distribution of zeta zeros and mention some of my related works on the subject.

**Date:**April 5, 2022

Video link

**Speaker:**Semih Ozlem (Yeditepe University, Turkey)

**Title:**

*On the motivic Galois group of a number field*

**Abstract :**Aim of this talk is to briefly introduce the motivic Galois group and state a potential answer to Langlands' conjecture regarding the relation of Langlands' group and motivic Galois group.

**Date:**March 15, 2022

Video link

**Speaker:**Cristian David Gonzalez Aviles (Universidad de La Serena, Chile)

**Title :**

*Totally singular algebraic groups*

**Abstract :**I will define the groups of the title and discuss some examples. When they are of positive dimension, these groups exist only over an imperfect field. We will see examples in dimension 1 related to the purely inseparable forms of the additive group studied by Russell in 1970. We will also see some examples of arbitrarily high dimensions. I hope to convince the audience that this class of algebraic groups is both interesting and quite large!

**Date:**February 1, 2022

Video link