Project Leader(s): Angelica Babei

Project Description:  TBD

Statement of Equity, Diversity and Inclusion: TBD

Project Leader(s): Antoine Leudière

Project Description:  Elliptic curves are among the most important objects in number theory. In particular, the Weil pairing for elliptic curves establishes a link between torsion points of elliptic curves, and roots of unity. The goal of this project is to give an efficient algorithm to compute the Weil pairing not on elliptic curves, but on Drinfeld modules.

Drinfeld modules are the function field analogue of elliptic curves, in the sense that, loosely speaking, the theory of Drinfeld modules resembles that of elliptic curves upon replacing all instances of the ring Z by the ring Fq[T]. Drinfeld modules were introduced in 1974 to solve the Langlands conjectures for linear groups of a function field, and subsequently became a fundamental tool in the arithmetic of function fields. Much later, Drinfeld modules were used for applications to computer algebra, or to coding theory and code-based cryptography.

Our goal will be to give a fast algorithm for the computation of the Weil pairing for Drinfeld modules, carefully analyze its complexity, and implement it.

The Weil pairing for Drinfeld modules was defined by van der Heiden in 2002, using Anderson motives. As it stands, this definition has no analogue for elliptic curves. In 2020, Katten gave explicit formulas for the Weil pairing, which yield fundamental insights, but are slow to compute.

It is not expected that participants already know the theory of Drinfeld modules. We will start by learning basic definitions on Drinfeld modules, before moving on to the state of the art on algorithms for Drinfeld modules. Then, we will aim at carefully understanding the works of van der Heiden and Katen, in an attempt to find a faster algorithm.

Statement of Equity, Diversity and Inclusion: Mathematics should be universal, yet our community too often fails at upholding that standard. Excellence is not exclusion, and I will do everything I can to ensure that all members of my group feel heard and valued. I want to build a space where people communicate, doubt, and explore with joy, and love.

In light of current events, as we watch the world lose any sense of humanity, I hope Rethinking Number Theory will stand as a refuge of peace and sanity. Some things are more important than science, but perhaps we can make mathematics be a unifying force, and not a divisive one.

Project Leader(s): Jeff Hatley and Zack Porat

Project Description:  Let us call two elliptic curves p-congruent if, for a prime p, their mod p Galois representations are isomorphic. When two elliptic curves are p-congruent, they “know” some interesting arithmetic information about each other; on the other hand, a famous conjecture of Frey and Mazur suggests that, for sufficiently large primes p, there are no non-isogenous p-congruent curves.

When the mod p representations are irreducible, p-congruence can be verified by computing an explicit, finite amount of data. This project will study, both theoretically and (especially) computationally, some of the statistical trends in this data, with the aim of uncovering some interesting new conjectures.

Statement of Equity, Diversity and Inclusion: Research in mathematics is a challenging yet worthwhile endeavor and we believe that it should be open to anyone who wishes to engage with it.  A strong mathematical community hinges on a wide range of perspectives; we aim to foster a space where all participants can be heard and can contribute on equal terms.  We affirm that every person’s value extends beyond their mathematical output.  Growth, curiosity, and collaboration matter just as much as any result.  RNT lowers barriers to participation in research, and we could not be happier to help bring together a diverse array of voices from the math community.  This project will be a collective learning experience, and we are so excited to work together with you to tackle some interesting questions! 

Project Leader(s): Melissa Emory and Tian An Wong

Project Description:  The explicit formulas of analytic number theory, going back to Riemann, relate sums over zeroes of L-functions in terms of sums over prime numbers. The 1972 reformulation by André Weil expresses the latter in terms of the Weil group, a dense subgroup of the absolute Galois group. In 2014, Arthur presented an explicit formula for automorphic L-functions in terms of the conjectural automorphic Langlands group, expected to be a certain extension of the Weil group. The goal of this project is to prove this formula. Succeeding this, we will also look for applications related to Sato-Tate conjectures, which describe the distribution of arithmetic objects such as Frobenius elements and Satake parameters.

Statement of Equity, Diversity and Inclusion: Our core philosophy is that mathematics is a human endeavor to which everyone can—and should—contribute. We believe that a community is strongest when it reflects a wide range of perspectives, backgrounds, and levels of experience. To illustrate this, we often look to the analogy of music: If only the world’s greatest pianist were allowed to play, most of us would never get to hear the beauty of piano music. Furthermore, a world where only one person ever played would be a silent and repetitive one. The same is true for mathematics. Our field thrives not because of a few "soloists," but because of the collective effort of a diverse community. While we all put in the effort and achieve varying results, the value lies in the process. By inviting everyone to the bench, we ensure that the "music" of mathematics remains vibrant, accessible, and evolving.

Our goal is to foster an inclusive environment where:

• Every voice is heard: We prioritize active listening and ensure that power dynamics do not silence emerging ideas.

• Meaningful contribution: Every team member, regardless of their starting point, should feel they have a clear and valued role in the research process.

• The Joy of Discovery: Beyond the technical results, we want every participant to experience the genuine joy of doing mathematics together.

Through insightful discussions and collaborative exploration, we look forward to uncovering the beauty of the Langlands program with our group members.

Project Leader(s): Ayla Gafni

Project Description:  A 2024 breakthrough of Guth and Maynard gave new estimates for large values of Dirichlet polynomials and new zero-density estimates for the Riemann zeta function.  These results have important implications toward understanding primes in short intervals.   In particular, these zero-density results lower the threshold for the length of interval in which the prime number theorem can be proven to hold always or almost always.   By studying Dirichlet L-functions instead of the Riemann zeta function, one can glean similar information about primes in arithmetic progressions.  This project will explore the implications of zero-density estimates on primes in arithmetic progressions in short intervals.  In cases where results are known for almost all intervals, we will work to find quantitative estimates for the exceptional set.

Statement of Equity, Diversity and Inclusion: At its heart, mathematics is a pursuit of collective human knowledge.   I have always believed that mathematics research is most productive when it is collaborative and not competitive.  It is vital that every voice feels welcome and included into the mathematical conversation.  Creating an inclusive and friendly research environment is the best way to inspire creativity and novel ideas.  Throughout my research career, I have had the privilege to collaborate with mathematicians from many different countries, backgrounds, and generations.  I value every perspective, and I look forward to joining the RNT community.

Project Leader(s): Wissam Ghantous

Project Description:  TBD

Statement of Equity, Diversity and Inclusion: TBD

Mentor: Chloe Stewart

Statement of Equity, Diversity and Inclusion: Diversity, equity, and inclusion are vital values to uphold, not only in mathematics, but in every aspect of life.  The bottom line is that every person deserves safety, respect, and to be supported in their goals and dreams.  When those needs are met people are able to accomplish amazing things, in our case innovative and joyful mathematics.  I think it’s important to say that creating an inclusive environment where everyone can reach their potential is not a simple matter.  It cannot be accomplished without an acknowledgment and understanding of current and historical oppression.

On a more personal note, I firmly believe that I would not be in math and doing what I love if my mentors had not been committed to the values of diversity, equity, and inclusion. Since I owe so much to the caring and dedication of my mentors, I try to give that support to others whenever I get the chance.  As part of that mission, I am extremely excited to contribute to the mission of RNT by supporting graduate student participants this year as the graduate student mentor.