BARD 6 Lightning Talks

Lightning above Tiger Stadium on the LSU campus. Photo by ESPN.

Lightning talks are short, informal talks on research or expository topics. The length of talks for BARD 6 will be 5 minutes. A list of speakers, with titles and abstracts, is posted below.

O. David Agbolade (Louisiana State University)

TBA

David Aiken (Louisiana State University)

TBA

Paresh Singh Arora (Louisiana State University)

TBA

Che-Wei Hsu (National Taiwan University)

Weierstrass mock modular forms associated with elliptic curves

Weierstrass mock modular forms are weight 0 mock modular forms associated with elliptic curves over $\mathbb{Q}$ . They were first observed by Guerzhoy and later studied by Alfes, Griffin, Ono, and Rolen. Theoretically, they can be constructed using the modified Weierstrass zeta-function, the Eichler integral, and canonical modular functions. Recently, I have been working on explicitly computing some special cases and exploring deeper underlying phenomena.

Ayanava Mandal (Louisiana State University)

Modular symbols and Eichler–Shimura theorems

Eichler–Shimura isomorphisms provide a bridge between analytic modular forms and the geometry of modular curves via cohomology. Modular symbols make this connection computationally explicit. In this talk, we will explain the modular symbols formalism, introduce the Eichler–Shimura theorems, and show how it leads to computation in Sage.

Julian Michele (University of Florida)

TBA

Esme Rosen (Louisiana State University)

The Explicit Hypergeometric Modularity Method Calculator

We provide a demonstration of the Explicit Hypergeometric Modularity Method (EHMM) introduced by Allen, Grove, Long, and Tu. The emphasis is on how computational tools in Sagemath related to the EHMM can be used in finding and proving connections between modular forms and hypergeometric motives.

Erick Ross (Clemson University)

Dimension sequences of modular forms

For $N \geq 1$ , let $S_{2}^{\text{new}}(N)$ denote the newspace of cuspidal modular forms of weight $2$ and level $N$ . In 2004, Greg Martin conjectured that as a sequence in $N$ , $\dim S_2^{\text{new}}(N)$ takes on all possible natural numbers. In this presentation, we investigate several generalizations and variations of this type of problem. In each case, we provide a complete characterization of when such a property holds.

Sayantan Santra (University of Oklahoma)

On evenness of coefficients of newforms

Using a construction by Deligne–Serre, we’re able to attach a linear Galois representation to a newform. This representation can then be used to study the congruence properties of the coefficients of that newform modulo some prime. In this talk, I’ll talk about some results obtained using this method, focusing on the prime 2.

Darren Schmidt (University of Florida)

Newton polygons and Ekedahl–Oort types of abelian covers of $\mathbb{P}^1$ branched at three points

We developed an algorithm to compute the Newton polygon and Ekedahl–Oort Type of every curve that is an abelian cover of $\mathbb{P}^1$ branched at three points of a given genus. We present data which shows that such curves with a Newton polygon or Ekedahl–Oort type that is believed to be unlikely, such as supersingular and superspecial curves, occur much more often than expected.

Iris Shi (University of Florida)

Artin–Schreier curves with minimal $a$ -number

Booher and Cais showed that the $a$ -number of an Artin–Schreier curve has an explicitly calculable lower bound. We provide families of curves attaining this lower bound and other results that help to show that the bound is sharp.