# NTSGrad Spring 2018/Abstracts

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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 We use conditions on the discriminant of an abelian extension $K/\mathbb{Q}$ to classify unramified extensions $L/K$ normal over $\mathbb{Q}$ where the (nontrivial) commutator subgroup of $\text{Gal}(L/\mathbb{Q})$ 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 $M/\mathbb{Q}( \sqrt{d})$ normal over $\mathbb{Q}$ with $\text{Gal}(M/\mathbb{Q}( \sqrt{d})) = H_8$ (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 $p_k | d_k$.

## Apr 10

 Niudun Wang Nodal Domains of Maass Forms Hecke-Maass cusp forms on modular surfaces produce nodal lines that divide the surface into disjoint nodal domains. I will briefly talk about this process and estimate the number of nodal domains as the eigenvalues vary.

## Apr 17

 Qiao He An Introduction to Automorphic Representations Automorphic representation is a powerful tool to study L-functions. For me, Tate's marvelous thesis is the real beginning of the whole theory. So I will start with Tate's thesis, which is really the automorphic representation of $GL_1$. Then I will talk about how to generalize Tate's idea to higher dimensions and explain some ideas behind Langlands program. If there is still time left, I will also mention the trace formula and use it to prove the classical Poisson summation formula.

## Apr 23

 Iván Ongay Valverde Definability of Frobenius Orbits and a Result on Rational Distance Sets In this talk I will present a paper by Héctor Pastén. We will talk about the meaning of definability in a ring and how having a formula that identifies Frobenius orbits can help you show an analogous case of Hilbert's tenth problem (the one asking for an algorithm that tells you if a diophantine equation is solvable or not). Finally, if time permits, we will do an application that solves the existence of a dense set in the plane with rational distances, assuming some form of the ABC conjecture. This last question was proposed by Erdös and Ulam.

## Apr 24

 Brandon Boggess Moving from Local to Global What do problems over local fields tell us about global problems?

## May 1

 Qiao He An Introduction to Automorphic Representations - Part II Last time I talked about Tate's thesis, which is actually the theory of automorphic representation of GL_1. This time I will continue. First, I will give the definition of automorphic representation, and use Hecke characters and modular forms to motivate the definition. Then I will explain some classical results about automorphic representation, and discuss how automorphic representations are related to L-functions.

## May 8

 Sun Woo Park Parametrization of elliptic curves by Shimura curves Let f be a weight-2 newform on $\Gamma_0(N)$. Given a fixed isogeny class of semistable elliptic curves over $\mathbb{Q}$, for some $N$ there exists a distinguished element $A$ of the isogeny class such that $A$ is the strong modular curve attached to f. In fact, $A$ is a quotient of $J_0(N)$ by an abelian variety, from which we can obtain a covering map $\pi: X_0(N) \rightarrow A$. Based on Ribet and Takahashi’s paper, I will discuss the properties of the covering map as well as its generalization to Shimura curves.