Project Leader: Alex Barrios and Manami Roy

Project Description: Given a rational elliptic curve, one can use Tate's algorithm to compute its conductor, which encodes local information of the elliptic curve at each prime. In addition, Tate's algorithm also outputs the Néron type and Tamagawa number at each prime, which are closely related to the number of irreducible components of the elliptic curve over finite fields. It is well known that isogenous elliptic curves have the same conductor. This project will investigate how the Néron types and Tamagawa numbers are related for isogenous elliptic curves. This question has been well studied in the literature but remains open for special cases corresponding to a 2- or 3-isogeny.

Using recent work, we will pursue this question by considering parameterized families of elliptic curves. We will first look at the case when there is a non-trivial torsion point and then build towards the general case. By studying these parameterized families, we aim to find the necessary and sufficient conditions on the parameters which give the Néron types and Tamagawa numbers for the isogenous elliptic curves.

Statement of Equity, Diversity and Inclusion: Mathematics is a fundamental subject for our scientific progression, but we are far from establishing an equal community within our subject. We dream of a future where we can achieve equity for everyone regardless of gender, nationality, race, religion, sexuality, and social-economic status. During this project, our goal is to create a welcoming and inclusive environment where we can all learn from each other; we believe that this is best accomplished by humanizing mathematics and sharing our personal experiences. Our project is designed to be accessible, with various entry points into the wondrous realm of the theory of elliptic curves. By having theoretical and computational aspects, we aim to allow all group members from various backgrounds to learn from each other and contribute in their own way. We will ensure that everyone's voice is heard.

Project Leader: Lea Beneish

Project Description: Lattices (and their theta functions) have a wide range of applications including sphere packings, quadratic forms, elliptic curves, L-functions, and string theory. The theory of codes, which was developed in order to detect possible transmission errors, can be viewed in analogy to the theory of lattices. The relation between the two has been well studied; there is a thorough dictionary between the terminology used to describe codes and lattices, given a binary linear code one can construct a lattice from that code, and there is a correspondence between (doubly even) self-dual binary linear codes of length n and certain (even) unimodular lattices in Rn. Vertex operator algebras (VOAs) were introduced to mathematics in order to understand the Monster group and the structure of the moonshine module. VOAs have since also appeared in the study of infinite-dimensional Lie algebras. In a sense, VOAs can also be viewed in analogy to lattices and given an even positive definite lattice one can construct its corresponding lattice vertex operator algebra.

Recent work towards characterizing these objects and, in certain cases, the sporadic simple groups which act naturally on them suggests a correspondence between vertex operator algebras, even positive definite lattices, and doubly-even linear binary codes. Our goal in this project will be to further develop this correspondence. We will first consider known examples of lattices and find the associated codes and vertex operator algebras. From there we may consider classifying these correspondences for certain types of lattices. We can also compare the theta functions of the lattices, so-called dual theta functions of the codes, and graded characters of the vertex operator algebras (known to be modular under certain nice conditions) and determine how to recover two from one of them. For specifically chosen examples of lattices we can ask if these correspondences give us any new applications.

Statement of Equity, Diversity and Inclusion: I believe mathematical potential is equally distributed across all groups. The opportunities and support required to actualize that potential are still unfortunately far from equal. The entire mathematical community stands to benefit from the promotion of equity, diversity, and inclusion. I am thrilled to be a part of “Rethinking Number Theory 2,” a program that promotes a comfortable and fun research experience for all of its participants. I am determined to make the working environment for this project a positive, welcoming, and inclusive one. I hope to cultivate a space where everyone is comfortable sharing their ideas and where everyone feels heard, respected, and appreciated by their groupmates.

Project Leader: Gil Moss

Project Description: In studying representations of the general linear group of a finite field, Ilya Piatetski-Shapiro discovered that there are certain invariants, called gamma factors, that can be attached to irreducible representations that characterize them uniquely. He wrote a short and elegant book explaining this, called "Complex representations of GL(2,K) for a finite field K." His motivation was that the representation theory of GL(2,K) for a finite field K controls a large part of the representation theory of GL(2,F) where F is a nonarchimedean local field, which, in turn, is essential for understanding automorphic forms. Some very deep number theory phenomena arise from congruences mod-ℓ between automorphic forms, where ℓ is a prime number, and this motivates studying the mod-ℓ representation theory of GL(2,K) for a finite field K of characteristic p. In other words replace the complex numbers by a field of positive characteristic ℓ. For most primes ℓ≠p, the theory looks exactly the same as for complex representations, but for some ℓ, new phenomena appear and new techniques are required. Our goal in this project will be to find a counterexample to Piatetski-Shapiro's theorem in the mod-ℓ situation, in other words, to find two non-isomorphic ℓ-modular representations of GL(2,K) that have the same gamma factors. I don't think this will be too hard. If we find a counterexample, we can then try to come up with an enhanced construction of gamma factors that recover Piatetski-Shapiro's theorem in the mod-ℓ case.

Statement of Equity, Diversity and Inclusion: I am committed to making this project and its working environment a welcoming and inclusive experience. We will struggle, be confused, and not know what we're doing, and that will be part of the process. I am committed to helping make the world of math research more diverse and equitable. I acknowledge and value the ways in which other people's backgrounds and experiences are different from my own. I'm looking forward to thinking through ways to change things to make number theory work for more people.

Project Leader: Angela Robinson

Project Description: There is a class of public-key cryptography based on linear error-correcting codes. The decoder used during error correction directly affects the security of a code-based cryptosystem because, often, the private key is used in the process of recovering a shared secret from a syndrome. Correlations between error patterns that lead to decoding failures and the private key of a scheme have been discovered, leading cryptographers to work diligently to minimize decoding failures. In this project, we will look closely at the iterative decoders used by cryptosystem BIKE, a 3rd round proposal in the NIST Post Quantum Cryptography standard process. The natures of these iterative, bit-flipping decoders make them difficult to analyze in a closed form. Cryptographers have thus relied on measurements of weakened versions of BIKE with a decryption failure rate that is high enough to measure with a reasonable amount of computational resources. These measurements have then been extrapolated to design parameters for BIKE that are believed to provide a sufficiently low decryption failure rate for cryptographic purposes.

A known problem with such extrapolations is that they don’t account for error floors caused by error vectors that are equidistant between two different codewords. Such error vectors cannot be recovered with better than 50% accuracy. While such error vectors are rare enough that they should not affect the extrapolations, it is unclear whether similar effects might occur with error vectors that are nearly equidistant between two different codewords. By examining the performance of the BIKE decoder on this class of error vectors, we will try to determine the accuracy of the extrapolated decryption failure analysis of the BIKE decoder. We will use a mix of probability theory and BIKE decoder implementations in Magma (or other software).

Statement of Equity, Diversity and Inclusion: I fully support the promotion of equity, diversity and inclusion in mathematics. I have devoted significant time and effort towards engaging with underrepresented groups in mathematics and building communities within mathematics to better support such groups. I established an AWM student chapter at Florida Atlantic University, served as a Women and Mathematics Postdoctoral Ambassador for the Institute of Advanced Studies, organized the inaugural FWIMD: Florida Women in Mathematics Day, and have been involved with the CryptograpHers Cybersecurity summer camp and the Girls Talk Math UMD summer camp specially designed for high school girls. I am committed to ensuring all participants are welcome, heard, and valued.

Project Leader: Soumya Sankar

Project Description: The p-torsion of an abelian variety in characteristic p is a mysterious object, rich in the amount of questions that can be asked about it. Several invariants associated with this object help us understand it from different angles. These include invariants such as p-ranks, a-numbers, Newton Polygons, etc., which give us both geometric and arithmetic information about abelian varieties. One big open question about these invariants is which of them occur for Jacobians of curves? Further, if some of them can occur, can we find examples thereof? How many times can a certain invariant occur, and what does that even mean? In this project, we will be exploring these questions in the context of specific families of curves that have extra structure (e.g. Artin-Schreier curves). The goal is to understand these invariants, learn about some big questions in the area and get some hands-on experience in computing examples and gathering data. While I approach such questions from the perspective of arithmetic geometry and arithmetic statistics, the project has several possible directions and parts that require little to no background in either.

Statement of Equity, Diversity and Inclusion: Mathematics, like every other human endeavor, benefits from multiple perspectives, multiple backgrounds, multiple stories. No matter what your story is, you can contribute to mathematics. My goal is to help build a space where you can do so. The aim of this project is not only to learn mathematics or how to approach it, but also to learn how to work with other people, learn from one another and support each other. Small, supportive spaces can be extremely helpful in both the mathematical and psychological development of a person and I hope that together, we can create one. I acknowledge different people come from different circumstances, think about mathematics differently, and I am fully committed to making this workshop a safe space for people to be themselves and think about mathematics the way they want to. I hope that I can contribute to making this workshop an inclusive and welcoming environment.

Project Leader: Tian An Wong

Professional Website: http://www-personal.umd.umich.edu/~tiananw/

Project Description: The classical Dedekind-Rademacher homomorphism is an integer-valued homomorphism on a congruence subgroup of SL(2,Z) of level p, whose values can be expressed in terms of Dedekind sums. It can be defined as the period of the associated Eisenstein series. Classically, it plays a role in Mazur's study of the Eisenstein ideal, Merel's study of Eisenstein elements in the homology of the modular curve, and more recently, Darmon-Vonk-Pozzi's study of rigid meromorphic cocycles on SL(2,Z[1/p]). In this project our goal is to attempt to construct the analogous homomorphism over imaginary quadratic fields, and in succeeding explore the circle of ideas that it relates to.

Statement of Equity, Diversity, and Inclusion: Number theory, despite its seemingly low barrier to entry as first courses in number theory and REUs suggest, can in fact be quite a hostile field. As the "queen of mathematics" it has attracted some of the most creative and clever minds, but also some of the most elitist and competitive ones. By the nature of power and privilege, it also reflects the broader tendency of institutional structures favoring those at the intersections of white, straight, able, rich, cis, male. That's why this workshop is awesome! Change is hard to make, especially within academia, so here's hoping that this small effort will grow a community of friendly number theorists! I have found these two posts by Isabel Laba to be particularly inspiring of late; they touch on how math research might be done differently.