NTSGrad Fall2023/Abstracts

This page contains the titles and abstracts for talks scheduled in the Fall 2023 semester. To go back to the main GNTS page for the semester, click here.

9/12

Joey Yu Luo

Geometric proof of Hurwitz class number relation

In this talk I will introduce the Hurwitz class number relation, and give a geometric proof using the modular curves over complex number. The main ingredients are different perspective of elliptic curves. First year graduate students who are interested in number theory are welcome.

9/26

Eiki Norizuki

Mass Formula

I will talk about a nice result by Serre which can be seen as counting the totally ramified extensions of a local field by an appropriate weight. By easy computations, one can arrive at analogous mass formulas for other extensions from Serre's mass formula. I will mention how it relates to other problems in number theory.

10/3

Caroline Nunn

Motivating class field theory

In this talk, I will give an outline of the main ideas of class field theory. I will begin by investigating the structure of the Galois group of an abelian extension of number fields using local information at unramified primes. I will then show how, in the case of cyclotomic fields, this local information can be pieced together to recover the full Galois group. This will lead us to the main results of class field theory. I will end with a number theoretic application to the problem of representing primes in the form x^2+ny^2.

10/17

Hyun Jong Kim

An integral big monodromy theorem

Associated to a family of curves C -> S are ell-adic monodromy representations, which generalize Galois representations. I will discuss part my ongoing thesis work demonstrating a big monodromy result for the moduli space of superelliptic curves. This result uses an arithmeticity result of reduced Burau representations of Venkataramana and clutching methods of Achter and Pries. Time permitting, I will also describe applications of this big monodromy result in other parts of my thesis --- it can be used to prove a Cohen-Lenstra result for function fields and to prove a result on the vanishing of zeta functions for Kummer curves over the projective line over finite fields.

10/31

Yifan Wei

rEaL geNuS onE cuRvE aNd seCtiOn conJecTure

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11/14

Yihan Gu

Introduction to Shimura Varieties

Shimura varieties are nice varieties having both Hecke symmetry and Galois Symmetry. In this talk, we will see how to define Shimura varieties, get some ideas where Hecke Symmetry and Galois Symmetry appear in Shimura varieties and some compactifications of Shimura varieties.

11/21

Tejasi Bhatnagar

Lifting abelian varieties from char p to char 0.

I’ll motivate a few ideas and give a broad picture of the theory of Serre and Tate, that is, the study of lifting (ordinary) abelian varieties from chap to char 0. In particular, I’ll also talk about how this connects to the study of moduli space of abelian varieties in char p and deformation theory.

11/28

Sun Woo Park

Random Matrix Model and Selmer groups

I will give a brief introduction to Poonen and Rain's heuristics on utilizing random matrix model to obtain a heuristic on the probability distribution of Selmer groups of twist families of elliptic curves over global fields. If time allows, we will try to investigate explicit examples when the heuristic does and does not work out.

12/5

John Yin

How the Number of Points of An Elliptic Curve Over a Fixed Prime Field Varies

The number of points on a fixed elliptic curve over a finite field with p elements is of the form p+1+a_p, where |a_p| < 2sqrt(p). Fixing p, one can consider the distribution of a_p as E varies over elliptic curves over Fp. I will present on details of this distribution.