# Bayou Arithmetic Research Days

Number theory mini-conferences in Louisiana

## BARD 1 Lightning Talks

Lightning talks are short, informal talks on research or expository topics. The length of talks for BARD 1 will be (up to) 10 minutes. Our speakers, in the order they will present, are:

Kalani Thalagoda (University of North Carolina Greensboro)

Computing Bianchi Modular Forms

Classical modular forms have a rich history of theorems and conjectures coming to life through extensive explicit computations. On the other hand, explicit examples of modular forms over imaginary quadratic fields are limited, due to complications that arise from the class group. My current research works toward filling in this gap. In this lightning talk, I will talk about what goes into computing modular forms and apply these techniques to prove the modularity of a particular elliptic curve.

Brian Grove (Louisiana State University)

Hypergeometric Moments

Moments for hypergeometric functions over finite fields have recently been introduced by Ono, Saikia, and Saad. I will introduce these moments and mention an idea for generalizing their work.

Prerna Agarwal (Louisiana State University)

Diving Deeper into Supercuspidal Representations

Reeder-Yu introduced certain low positive depth supercuspidal representations of $p$-adic groups called the epipelagic representations. These representations generalize the simple supercuspidal representations of Gross-Reeder, which have the lowest possible depth. Epipelagic representations also arise in recent work on the Langlands correspondence; for example, simple supercuspidals arise in the automorphic data corresponding to the Kloosterman $l$-adic sheaf. In this talk, we talk about a construction of these supercuspidal representations.

Andrea Bourque (Louisiana State University)

Geometry from Partitions

There is a family of geometric objects called Schubert varieties, which are indexed by partitions. Furthermore, geometric information of these spaces, such as dimension and singular locus, can be described just by looking at the corresponding Young diagram. The goal of the talk is to introduce these facts and inspire number theorists to think about partitions in a new way.

Matthias Storzer (Max Planck Institute)

Modularity of Nahm Sums

We will discuss recent progress on a conjecture made by Werner Nahm and Don Zagier concerning the modularity of certain q-hypergemoetric series, so-called Nahm sums.

Emma Lien (Louisiana State University)

Quadratic Forms and Ray Class Groups

A classical result in algebraic number theory is that there is a 1-to-1 correspondence between the ideal class group of an imaginary quadratic extension $K$ of discriminant $d$ and primitive binary quadratic forms of discriminant $d$, i.e., binary quadratic forms of discriminant $d$ up to $\text{SL}_2(\mathbb{Z})$ equivalence. In a paper by Ick Sun Eum, Ja Kyung Koo, and Dong Hwa Shin, they generalize this to give a bijection between ray class groups of modulus $\mathfrak{n} = N\mathcal{O}_K$ and binary quadratic forms up to $\pm\Gamma_1(N)$ equivalence. We’ll explore this result in detail and discuss its consequences.

Evangelos Nastas (State University of New York)

On generalizing a bound of D.R. Heath-Brown for the cubic Weyl sum

In this lightning talk, bounds for the cubic Weyl sum, especially a bound by D.R. Heath-Brown for the cubic Weyl sum will be introduced. This will be followed by exploring ideas for its generalization, notably the quite involved estimation of exponential sums with prime power moduli, to attain an estimate for the cubic Weyl sum with fewer conditions.

Pranabesh Das (Xavier University of Louisiana)

An Application of Multi-Frey-Hellegouarch Approach

Let $k \geq 1$, $n \geq 2$ be integers. A power sum is a sum of the form $x_1^k + x_2^k + \cdots + x_n^k$ where $x_1, x_2, \ldots, x_n$ are all integers. Perfect powers appearing in power sums have been well studied in the literature and are an active field of research. In this short talk, we consider the Diophantine equation $(x-r)^5 + x^5 + (x+r)^5 = y^n \ (n \geq 2)$, where $r, x, y, \in \mathbb{Z}$ and $r$ is composed of certain fixed primes. We determine the integral solutions of this Diophantine equation as an application of modularity using the multi-Frey-Hellegouarch method. This is a joint work with Dey, Koutsianas, and Tzanakis.