NTSGrad Spring 2022/Abstracts

This page contains the titles and abstracts for talks scheduled in the Spring 2022 semester. To go back to the main GNTS page, click here.

Jan 25

Jerry Yu Fu

Canonical lifting and isogeny classes of Abelian varieties over finite field

I will give a brief introduction from Serre-Tate's canonical lifting, the Grothendieck-Messing theory and their applications to class group and estimation of size of isogeny classes of certain type of abelian varieties over finite fields. I will present some recently proved results by me and some with my collaborator.

Feb 1

TBA

Feb 8

Di Chen

A non-trivial bound on 5-torsion in class groups.

I will discuss A. Shankar and J. Tsimerman’s recent work on a non-trivial bound on 5-torsion in class groups of imaginary quadratic fields. I focus on ideas of proofs and assume several black boxes without proofs. This is a good application of elliptic curves and Galois cohomology.

Feb 15

TBA

Feb 25

TBA

Mar 1

TBA

Mar 8

TBA

Mar 15

TBA

Mar 22

TBA

Mar 29

TBA

Apr 5

TBA

I will talk about local reciprocity, a correspondence of the Galois group of the maximal abelian extension and the multiplicative group. In particular, I will talk about Lubin-Tate theory which constructs this map.

Apr 12

TBA

In this talk, we will walk through a simple example of counting quadratic extensions using the discriminant. Although, this has been done using classical methods, we will highlight the techniques used by Bhargava through our example, that were essentially used to count the higher degree cases.

Apr 19

TBA

Given a positive definite quadratic form X_1^2+...+X_n^2, a natural question to ask is can we find a formula for $r_n(m)=\#\{X\in Z^n| Q(X)=m\}$. Although no explicit formula for $r_n(m)$ is known in general, there do exist an average formula, which is a prototype of the so called Siegel-Weil formula. In this talk, I will introduce Siegel-Weil formula, and show how Deuring's mass formula for supersingular elliptic curve and Hurwitz class number formula follows from Siegel-Weil formula.

Apr 26

TBA

I will motivate and introduce the definition of a height. Then, I will talk a bit about Arakelov height. This will then lead into a recent paper by Ellenberg, Satriano, and Zureick-Brown, which introduces a notion of height on stacks.

May 3

Jerry Yu Fu

Canonical lifting and size of isogeny classes

I will give a brief review from Serre-Tate's canonical lifting theorem, the Grothendieck-Messing theory and their applications to class group and isogeny classes of certain type of abelian varieties over finite fields. I will present some recently proved results by me and some with my collaborator.