NTSGrad Spring 2018/Abstracts

This page contains the titles and abstracts for talks scheduled in the Spring 2018 semester. To go back to the main NTSGrad page, click here.

Jan 23

Solly Parenti

Rankin-Selberg L-functions

What do you get when you cross an Eisenstein series with a cuspform? An L-function! Since there's no modular forms course this semester, I will try to squeeze in an entire semester's course on modular forms during the first part of this talk, and then I'll explain the Rankin-Selberg method of establishing analytic continuation of certain L-functions.

Jan 30

Wanlin Li

Intersection Theory on Modular Curves

My talk is based on the paper by François Charles with title "FROBENIUS DISTRIBUTION FOR PAIRS OF ELLIPTIC CURVES AND EXCEPTIONAL ISOGENIES". I will talk about the main theorem and give some intuition and heuristic behind it. I will also give a sketch of the proof.

Feb 6

Dongxi Ye

Modular Forms, Borcherds Lifting and Gross-Zagier Type CM Value Formulas

During the course of past decades, modular forms and Borcherds lifting have been playing an increasingly central role in number theory. In this talk, I will partially justify these by discussing some recent progress on some topics in number theory, such as representations by quadratic forms and Gross-Zagier type CM value formulas.

Feb 20

Ewan Dalby

The Cuspidal Rational Torsion Subgroup of J_0(p)

I will define the cuspidal rational torsion subgroup for the Jacobian of the modular curve J_0(N) and try to convince you that in the case of J_0(p) it is cyclic of order (p-1)/gcd(p-1,12).

Feb 27

Brandon Alberts

A Brief Introduction to Iwasawa Theory

A bare bones introduction to the subject of Iwasawa theory, its main results, and some of the tools used to prove them. This talk will serve as both a small taste of the subject and a prep talk for the upcoming Arizona Winter School.

Mar 13

Solly Parenti

Do You Even Lift?

Theta series are generating functions of the number of ways integers can be represented by quadratic forms. Using theta series, we will construct the theta lift as a way to transfer modular(ish) forms between groups.

Mar 20

Soumya Sankar

Finite Hypergeometric Functions: An Introduction

Finite Hypergeometric functions are finite field analogues of classical hypergeometric functions that come up in analysis. I will define these and talk about some ways in which they are useful in studying important number theoretic questions.

Apr 3

Brandon Alberts

Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations

This is a practice talk for a 20 minute presentation I will be giving at an AMS sectional. Feedback and constructive criticism are especially welcome. We study solutions to the Brauer embedding problem with restricted ramification. More specifically, suppose [math]\displaystyle{ G }[/math]and [math]\displaystyle{ A }[/math] are finite abelian groups, [math]\displaystyle{ E }[/math] is a central extension of [math]\displaystyle{ G }[/math] by [math]\displaystyle{ A }[/math], and [math]\displaystyle{ f:\textnormal{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow G\lt math\gt Insert formula here }[/math]</math> is a continuous homomorphism. We determine conditions on the discriminant of [math]\displaystyle{ f }[/math] that are equivalent to the existence of an unramified lift [math]\displaystyle{ \widetilde{f}:\textnormal{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow E }[/math] of [math]\displaystyle{ f }[/math]. As a consequence, we use conditions on the discriminant of an abelian extension $K/Q$ to classify unramified extensions $L/K$ normal over $\mathbb{Q}$ where the (nontrivial) commutator subgroup of $\textnormal{Gal}(L/\mathbb{Q})$ is contained in its center. This generalizes a result due to Lemmermeyer stating that the quadratic field of discriminant $d$, $\mathbb{Q}( \sqrt{d})$, has an unramified extension $M/Q( \sqrt{d})$ normal over $\mathbb{Q}$ with $\textnormal{Gal}(M/Q( \sqrt{d})) = H_8$ (the quaternion group) if and only if the discriminant factors $d = d_1 d_2 d_3$ into a product of three coprime discriminants, at most one of which is negative, satisfying $\left(\frac{d_i d_j}{p_k}\right) = 1 for each choice of $\{i, j, k\} = \{1, 2, 3\}$ and prime $p_k | d_k$.