Project Leader: Kwangho Choiy

Project Description: An infinite set of equivalence classes of irreducible smooth representations of p-adic groups is conjectured to be partitioned into finite sets, so-called L-packets, indexed by Langlands parameters which are homomorphisms of Weil-Deligne group to Langlands dual group of the p-adic group. This prediction has been proven in many cases over decades. The goal of this project is to explore applications of those known results to mysterious but tangible cases of lower rank groups and seek an explicit description of the internal structure of their L-packets. One interesting example is the case of SL2(Qp), whose L-packets are parametrized by irreducible representations of subgroups of the Klein 4-group. The project would further expect to strengthen knowledge and encourage inquisitiveness about fundamentals of the Langlands program.

Statement of Equity, Diversity and Inclusion: There is no boundary in pure joy of mathematics. We shall enjoy discussing and learning in the atmosphere which is full of involvement and encouragement. My hope is to cultivate each diverse idea and welcome different ways of thinking, as there are a variety of approaches and parts that anyone can contribute. It is supported that we all listen to every single voice and create an open collaborative space.

Project Leader: John Cullinan

Project Description: This aim of this project is to generalize to genus-2 hyperelliptic Jacobians the following result in the arithmetic of elliptic curves.

Let E and E' be isogenous elliptic curves over a finite field k. Since the curves are isogenous, the cardinalities |E(k)| and |E'(k)| are the same. However, the group structures E(k) and E'(k) need not be isomorphic.

Suppose E(k) and E'(k) are isomorphic as groups. What can be said about the groups E(K) and E'(K), as K varies over all finite extensions of k? Based on a number of previous works, one can show that:

• if the degree of the isogeny is an odd prime l, then if E(k) and E'(k) are isomorphic, and the cardinalities |E(k)| and |E'(k)| are divisible by l, then the groups E(K) and E'(K) are isomorphic for all finite extensions K/k.

• if the degree of the isogeny is 2, and K is the unique quadratic extension of k, then if E(k) and E'(k) are isomorphic as groups *and* if E(K) and E'(K) are isomorphic as groups, then E(L) and E'(L) are isomorphic as groups, for all finite extensions L/k.

Now suppose E and E' are defined over the rational numbers Q, and are also 2-isogenous over Q. Let's perform the following experiment. Call a prime p special if E(F_p) and E'(F_p) are isomorphic as groups, but E(F_{p^2}) and E'(F_{p^2}) are not. Can we compute the proportion of special primes?

In fact, this proportion does exist and we can compute it using matrix properties of the 2-adic representation attached to E and E' in conjunction with the known properties of special primes on elliptic curves.

Now consider the case where C and C' are genus 2 hyperelliptic curves with Jacobians J and J'. In this project we will first attempt to determine explicit criteria for J(K) and J'(K) to be isomorphic in finite fields K, generalizing the known literature on elliptic curves. If we can successfully do this, then we can turn to the question of proportion of special primes for isogenous Jacobians over Q.

Statement of Equity, Diversity and Inclusion: I am thrilled to be part of the third Rethinking Number Theory program. As a project leader, I am fully committed to creating an inclusive work environment where everyone feels comfortable, not only to contribute, but to thrive. We do our best learning when we are free to ask questions and admit what we do not understand. My hope is that we will practice asking questions of each other (including me!) and bring that bravery and freedom of inquiry beyond this workshop and into our regular working lives.

Number theory has ancient roots but, like most areas of mathematics, has applications that impact our present lives. Some of these applications (for example, to surveillance and cryptology) are actively harmful to innocent people in the United States and abroad. As educators, we must examine our role in this field and how we communicate the broader impacts of our work to our students. I hope that during this workshop we can take time out each day to discuss how we can talk about all of this to our students in the classroom, to our fellow faculty, and, more broadly, to the community at large.

Project Leader: Maria Fox

Project Description: Given an elliptic curve E over the complex numbers, the group of N-torsion points, E[N](C), is always isomorphic to Z/NZ x Z/NZ. Something similar happens if we study the N torsion in characteristic p, as long as p does not divide N: if E is any elliptic curve over \bar{F_p}, and p does not divide N, then E[N](\bar{F_p}) is isomorphic to Z/NZ x Z/NZ.

However, the p-torsion in characteristic p behaves very differently! Given an elliptic curve E over \bar{F_p}, the group E[p](\bar{F_p}) can be isomorphic to the group Z/pZ, or can consist of only the identity element. We say that two elliptic curves E_1 and E_2 over \bar{F_p} are in the same Ekedahl-Oort stratum if their p-torsion groups E_1[p](\bar{F_p}) and E_2[p](\bar{F_p}) are isomorphic. So, the space of all elliptic curves over \bar{F_p} consists of two Ekedahl-Oort strata: one stratum consisting of the collection of all elliptic curves with E[p](\bar{F_p}) congruent to Z/pZ and the other consisting of those elliptic curves with trivial p-torsion group E[p](\bar{F_p}).

Abelian varieties are higher-dimensional analogues of elliptic curves, and they are often studied with important extra structure, like a fixed set of symmetries. So, we can generalize the above picture by replacing the space of elliptic curves over \bar{F_p} with a space of abelian varieties with extra structure over \bar{F_p}. There is a similar notion of the Ekedahl-Oort stratification: two abelian varieties A_1 and and A_2 over \bar{F_p} are in the same Ekedahl-Oort stratum if their p-torsion groups A_1[p] and A_2[p] (equipped with extra symmetries) are isomorphic.

In this project, we'll study a particular space of abelian varieties in characteristic p (called a unitary Shimura variety), and seek to understand various properties of the Ekedahl-Oort strata: How many strata are there? What is the dimension of each stratum? How do these strata interact with other important geometric aspects of our space of abelian varieties?

Statement of Equity, Diversity and Inclusion: In order to build a kinder, stronger, and more productive mathematical community, I believe we should work towards a greater diversity of participants and perspectives, as well as more equitable access to resources and opportunities. The Rethinking Number Theory conference will give us a unique opportunity to reflect on ways to improve our community, and I look forward to hearing the ideas of my fellow participants.

As a project leader, I am committed to fostering an inclusive and supportive environment for all team members. I hope that everyone will feel empowered to share their questions, their thoughts, and their ideas. Though this is a number-theoretic project, the study of Ekedahl-Oort strata has both conceptual and computational aspects and involves ideas from linear algebra, representation theory, combinatorics, and algebraic geometry. I intend for this project to serve as a venue where we can bring together team members with a variety of interests and backgrounds. I hope we will learn a lot from each other!

Statement of Equity, Diversity and Inclusion: I am very honored to be a project leader for the third iteration of RNT. I am a firm believer that there is no right way or order in which to learn mathematics. I believe that with enough time, support, patience and desire, anyone is capable of contributing to any mathematical area. Furthermore, I believe diverse backgrounds and perspectives make mathematics and its environment richer. Unfortunately, many social constructs put barriers between persons of certain groups and these four basic criteria.

Over the last three thousand years, almost exclusively white men have driven the development of western mathematics. As a result, straight, white, cisgender, able men in math find themselves belonging to a rich, well-established culture. Forming such a culture for those outside of this dominant group requires care and intention. I believe that forming a welcoming community where all voices are heard and respected is a necessary step towards equity in mathematics. I look forward to working with a group of mathematicians whose intent is to learn something new not just about mathematics but also about how to best support those who are not always given a voice within it.

Project Leader: Hiram H. López Valdez and Welington Santos

Project Description: A linear code is the main object of study in coding theory; it supports reliable communication in classical and quantum communication, protects information through code-based cryptography, and helps to distribute data in storage systems. The automorphism group of a linear code is the subgroup of the group of monomial matrices which preserves the linear code; it is beneficial to determine the structure of the code, compute the weight distribution, and describe decoding algorithms.

The generalized Reed-Solomon codes are evaluation codes that depend on a set of points of a finite field and polynomials up to a certain degree. They are one of the most popular and fundamental families of linear codes because of their algebraic connections that allow the development of essential tools, such as a fast decoding algorithm.

Let C be a tensor product of generalized Reed-Solomon codes. This project will seek to understand the groups of automorphisms of C and the dual of C.

Statement of Equity, Diversity and Inclusion: We will provide a welcoming and inclusive environment for each participant. The project will be accessible to people who have different levels of training in mathematics, aiming that all interested parties can get involved and have space to contribute. We do not assume research experience and a background in coding theory, but we anticipate that people want to learn, work, and participate. By the end of this project, we expect at least the following outputs: participants will learn the basic steps to do research and be familiar with coding theory and automorphisms.

Project Leader: Amita Malik

Project Description: Problems of counting nature, especially primes of some certain type, lie at the heart of analytic number theory. Our RNT-3 project will focus on the number of extremal primes not exceeding some given large real number. These are the primes at the extreme ends of the Sato-Tate measure for elliptic curves. Though the asymptotics for these primes have been proved for the elliptic curves with complex multiplication (CM), the conjecture is wide open in the case of non-CM elliptic curves. Indeed, it is not even known if there are infinitely many such primes. Moreover, the current known upper bounds for these primes are far from the expected quantity. The project will have both algebraic and analytic flavors, however, it draws main inspiration from the analytic side.

Statement of Equity, Diversity and Inclusion: The idea is to explore the theory of extremal primes in an inclusive and supportive environment. Having participated in a Women in Numbers workshop and numerous AWM events, I myself have greatly benefited. I am committed to pay it forward and offer a research space where we all, including myself, are able to ask/answer questions without any fear of being judged and believe in equity. My hope is that by the end of the program, we all would have not just new great collaborators but also enlarged our respective circles of mathematicians we can trust for guidance and support when needed. Hopefully, we all would become just a little kinder, a little more empathetic as I believe it's never too late and never too much of that in our small number theory community, in particular. I am looking forward to being a part of the RNT-3 diverse family.