NTSGrad Spring 2022/Abstracts

This page contains the titles and abstracts for talks scheduled in the Fall 2021 semester. To go back to the main GNTS page, click here.

Jan 25

TBA

In his notes for students, Jordan has a list of general topics and references in number theory/algebraic geometry/arithmetic geometry that students in arithmetic geometry should be comfortable with after a certain point of time. I will introduce some language used in these general topics for beginners.

Feb 1

TBA

In this talk, I will go over the history of rational/integral points on curves. In particular, I will introduce a recent proof of the S-unit equation using p-adic period maps, given by Lawrence-Venkatesh.

Feb 8

TBA

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Feb 15

TBA

Geometry over char p is fascinating or frustrating, depending on who you are. However varieties over char 0 could be enjoyed by geometers of all kinds. We will dicuss one way of lifting a smooth projective variety from char p to char 0. After applying our technique to curves we briefly mention the situation in higher dimensions. And if time permits, we discuss a non-liftable example by Serre.

Feb 25

TBA

Mar 1

TBA

Mar 8

TBA

I will introduce Euler's classical result over Q, Klingen-Siegel theorem over totally real number fields, and Zagier's theorems and conjectures over general number fields. I will give many examples and discuss their proofs. If time permits, I will discuss its relation with K-theory.

Mar 15

TBA

Diophantine approximation is a crucial tool in studying integral points and Schlickewei's theorem is a very useful theorem in proving finiteness theorems on integral points. In the first part of my talk I will show some elegant proof as applications of the subspace theorem such as Vojta's theorem, the S-unit equation, and then I will introduce main conjectures: Vojta, Mordell, Bombieri and Lang, and their relations to each other.

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.