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

John Yin

Bertini Theorems over Finite Fields/Poonen Sieve

Consider the question: What's the probability that a projective plane curve of degree d over F_q is smooth as d approaches infinity? Assuming some sort of independence, this should be something like the product over closed points in P^2 of the proportion of plane curves which are smooth at the closed point. A version of this turns out to be true, and it is proven through the Poonen Sieve.

Feb 25

TBA

Mar 1

TBA

Mar 8

TBA

Mar 15

TBA

Mar 22

TBA

Mar 29

Tejasi Bhatnagar

The theorem of Honda and Tate

In this talk, we aim to understand the classification of abelian varieties over finite fields, up to isogeny. To every abelian variety, we can associate a certain algebraic number. This is called a Weil-q number. The Theorem of Honda and Tate tells us that, up to isogeny this association is a bijection. We won’t necessarily prove the entire theorem, but we will see bits and pieces to understand whatever we can and mostly try and get an understanding of the important objects that we’ll come across.

Apr 5

Hyun Jong Kim

Cohen-Lenstra for imaginary quadratic function fields

A Cohen-Lenstra type statement is one which claims or states that certain objects are distributed inversely proportional to the size of their automorphism groups. Originally stated for class groups of quadratic number fields, Ellenberg, Venkatesh, and Westerland showed that an analogue for imaginary quadratic function fields over finite fields hold. I will introduce the Cohen-Lenstra heuristics for number fields and outline the proof to Ellenberg, Venkatesh, and Westerland's theorem.

Apr 12

Qiao He

Does rigid analytic varieties has Hodge symmetry?

I will survey a paper by our last week’s speaker Alexander Petrov. It is well known that there is a symmetry between the Hodge numbers of a Kahler Manifold (in particular, projective variety). In the p-adic world, we have similar analytic space: rigid analytic variety. Then it is a natural question to ask whether the Hodge number of rigid analytic variety still has a symmetry. It turns out that the answer is no! I will try to explain how to construct a counterexample in the talk.

Apr 19

TBA

Apr 26

TBA

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.