Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting 2021

In the third year of the project, the scope of our activities and the number of people involved continue to grow. Our monthly meetings involve 25 researchers, 15 of whom are research scientists hired to directly support the goals of the project.

Notable accomplishments include:

• the creation of the site https://researchseminars.org which has become the standard source of information for online mathematical conferences and seminars worldwide — it builds on database expertise cultivated in our core research projects;
• continuing growth in the functionality and scope of the L-functions and Modular Forms Database https://www.lmfdb.org, culminating in the release of version 1.2 in October;
• the online Workshop on Arithmetic Geometry, Number Theory, and Computation at ICERM, focused around nine group projects involving more than 70 researchers both from within the Collaboration and around the world.

Specific results and achievements of PIs include:

• Jennifer Balakrishnan was a featured speaker at the Arizona Winter School, the highest profile number theory event in the US. She offered a mini-course Computational tools for quadratic Chabauty.
• Noam Elkies, with Klagsbrun, contributed New rank records for elliptic curves having rational torsion covering curves over \(\mathbb{Q}\) with torsion groups \(\mathbb{Z}/n\mathbb{Z}\) for \(n=2,3,4,6,\) and \(7\).
• Bjorn Poonen, with Kedlaya, Kolpakov, and Rubinstein, classified all sets of nonzero vectors in \(\mathbb{R}^3\) such that the angle formed by each pair is a rational multiple of \(\pi\). This includes a characterization of tetrahedra with rational dihedral angles, solving a 1976 problem of Conway and Jones. Their preprint, Space vectors forming rational angles, is a tour de force of computational arithmetic, including the solution of a polynomial of degree six with 105 terms in roots of unity.
• Andrew Sutherland, with Booker, found the first integer solution to the equation $$x^3+y^3+z^3 = 42$$ and new integer solutions to $$x^3+y^3+z^3 = 3.$$ This resolves a 1953 question of Mordell and completes a search begun by Miller and Woollett in 1954. Their manuscript, On a question of Mordell, presents a new algorithm to search for representations of positive integers as sums of three cubes, implemented via a massive parallel computation.
• In Counting elliptic curves with an isogeny of degree three with Pizzo and Pomerance and On a probabilistic local-global principle for torsion on elliptic curves with Cullinan and Kenney, John Voight has provided precise asymptotic counts of elliptic curves equipped with extra level structure.

The Collaboration website https://simonscollab.icerm.brown.edu/ lists dozens of preprints written by Collaboration members over this year. We are preparing a volume presenting key algorithms, examples, and foundational results supporting our work, with 21 submissions totaling over 450 pages.

Tetrahedra: From Aristotle’s Mistake to Unsolved Problems

Bjorn Poonen, Ph.D.
Massachusetts Institute of Technology

Tetrahedra are three-dimensional shapes with four triangular faces. Which tetrahedra can tile to fill a three-dimensional space? Which tetrahedra have rational dihedral angles (the angle between two intersecting planes)? Which tetrahedra can be sliced and reassembled into a cube? Each of these three problems has been around for at least 45 years, and one of them is over 2300 years old. In this lecture, Bjorn Poonen will discuss the status of these problems and explain how he solved one of them in collaboration with K. Kedlaya, A. Kolpakov, and M. Rubinstein.

Brendan Hassett, Collaboration Director
with J. Balakrishnan, N. Elkies, B. Poonen, A. Sutherland, and J. Voight

Arithmetic Geometry, Number Theory, and Computation:
Simons Collaboration Overview

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Brendan Hassett
in collaboration with J. Balakrishnan, N. Elkies, B. Poonen, A. Sutherland, and J. Voight

Arithmetic Geometry, Number Theory, and Computation:
Simons Collaboration Director’s Overview

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Jennifer Balakrishnan
Boston University

Rational Points on Modular Curves and Quadratic Chabauty

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The quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk) can be used to determine rational points on certain curves whose Jacobians have large Mordell–Weil rank. I will discuss some algorithmic aspects and highlight several recent successes of quadratic Chabauty, from the “cursed” modular curve and beyond.

John Voight
Dartmouth College

The L-functions and Modular Forms DataBase (LMFDB)

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The Langlands program is a set of conjectures that lie in deep theories of mathematical symmetry, connecting numerous subfields of mathematics. Recently, it has become feasible to do large scale computational verification of the predictions of the Langlands program, to test conjectures in higher-dimensional cases, and in particular to present the results in a way that is widely accessible. To this end, the L-functions and Modular Forms DataBase (LMFDB, https://www.lmfdb.org) was created to connect and organize the work of many mathematicians working broadly in this area. In this talk, we will first highlight the contributions made by the Simons Collaboration in advancing the LMFDB and then we will discuss possible future directions.