NTSGrad Fall 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.

9/13

Ivan Aidun

A Case Study in the Analogy Between Z and F_q[t]

An influential concept in modern number theory is the idea that the integers Z and the ring of polynomials over a finite field F_q[t] share many traits.  In this talk, I will discuss some particular examples of how this analogy can work, focusing on zeta functions and counting problems.  No prior familiarity will be required!

9/20

Jiaqi Hou

Poincare series and Petersson trace formula

I will talk about the Poincare series, which are basic examples of modular forms, and the Petersson trace formula for SL(2,Z). Then I will discuss some applications of Petersson's formula.

9/27

No speaker

10/4

Eiki Norizuki

p-adic L-functions

In this talk, I will look at how p-adic L-functions are constructed as first demonstrated by Kubota and Leopoldt. These are p-adic analogues of the Dirichlet L-functions and the main idea of the construction comes from interpolating the negative integer values of the classical L-functions. This talk should be accessible to everyone.

10/11

Sun Woo Park

Rank 2 local systems and Elliptic Curves

We'll understand some key properties of elliptic curves (Weil Pairing, eigenvalues of Frobenius, and poles of j-invariants) and try to see how these properties are closely tied in with understanding certain properties of rank 2 local systems over an open subset of the projective line $\mathbb{P}^1$.This is a preparation talk for the NTS talk on Thursday.

10/18

John Yin

Some Examples in Etale Cohomology

I will motivate, define, and give explicit examples of etale cohomology. In addition, I will compute the Galois action on etale cohomology in certain cases.

10/25

Sun Woo Park

2-Selmer groups and Markov Chains

This is a survey talk on Klagsbrun, Mazur, and Rubin’s approach on utilizing Markov chains to compute the upper bound of the distribution of ranks of quadratic twist families of elliptic curves over number fields. Though we won’t be able to go through all the details, we will try to first identify how one can use 2-Selmer groups to bound the rank of certain families of elliptic curves, and why Markov chains are relevant for understanding the distribution of 2-Selmer groups.

11/1

Stephen Waigner

The Circle Method: Part 1

I plan to briefly mention Waring's problem that is writing an integer as a sum of kth powers of other integers, and then go into a lot of detail on writing an integer as a sum of squares. Finally, I would like too say a few words about writing an odd integer as a sum of three primes and why the method fails in trying to write an even integer as a sum of two primes.

11/8

TBA

TBA

11/15

Tejasi Bhatnagar

Working with abelian varieties in characteristic p

I will try and motivate a lot of objects that we come across when we work with abelian varieties in characteristic p. For example, I plan to introduce the study of  (the moduli space of) abelian varieties in char p using   “stratifications” and “deformation theory”. As we talk broadly about these topics, we will also discuss what sorts of objects do we study and in general motivate ideas that we come across in char p.

11/22

Hyun Jong Kim

Points on certain Hurwitz schemes correspond to surjections from Jacobians of curves

I will introduce Hurwitz spaces to partially explain how Ellenberg-Venkatesh-Westerland proved a Cohen-Lenstra result for function fields and how Ellenberg-Li-Shusterman bounded the proportion of hyperelliptic zeta functions that vanish at fixed numbers.

11/29

Yifan Wei

A Chebotarev density theorem over p-adics

We show that for generically finite map between smooth projective varieties over Zp, the density of the image carries more information than the classical Chebotarev over finite fields. More interestingly, a mysterious symmetry shows up when you try to replace the prime number p by its inverse 1/p. We show that this symmetry partially follows from Poincare duality, and maybe explore more phenomenon of the sort. This is a joint work with John Yin and Asvin G.

12/6

Qiao He

A glimpse of unlikely intersection

I will motivate the problems considered in the field of unlikely intersection theory, including Mumford-Tate conjecture, Mordell-Lang conjecture and Andre-Oort conjecture. If time permits, I will give a sketch of a proof of Andre-Oort for product of modular curves.

12/13

Yu LUO

On the integral model of modular curve

I will start with some basic definition and properties of the modular forms and modular curves and its relation with the Langlands correspondence. Once this is done, I will talk about the integral model of modular curves and focus on the problems arising from point counting approach to the Langlands program.