Simons Collaborationon Arithmetic Geometry, Number Theory, and Computation

Help for those hosting virtual events

Andrew Sutherland, one of the research scientists working at MIT as part of the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation, recently co-hosted a virtual panel discussion, “Mathematics research online: hosting virtual events.” The panelists were made up of conference organizers who had early experience pivoting in-person talks to a virtual platform at the start of the COVID-19 pandemic. A video recording of the panel discussion, as well as Q&A and poll results collected during the session, are all available for review.

New Solution for the Sum of Three Cubes for 3!

It’s easy to represent 3 is a sum of three cubes 3 = 1³ + 1³ + 1³, but can we characterize all the solutions of x³ + y³ + z³ = 3 with x, y, and z integers?

Until this week, the only other solution known was 4³ + 4³ + (-5)³ = 3.

Andrew Booker and collaboration PI Andrew Sutherland just found the next simplest solution

3 = 569936821221962380720³ + (-569936821113563493509)³ + (-472715493453327032)³

Read more about it:

• Bristol University press release: http://www.bristol.ac.uk/maths/news/2019/number-3.html
• AMS blog post: https://blogs.ams.org/beyondreviews/2019/09/18/3/#more-2526
• Quanta Magazine article: https://www.quantamagazine.org/why-the-sum-of-three-cubes-is-a-hard-math-problem-20191105/

Sum of Three Cubes for 42 Finally Solved!

A team led by the University of Bristol and Massachusetts Institute of Technology (MIT) has solved the final piece of the famous 65-year-old maths puzzle with an answer for the most elusive number of all – 42.

The mathematical content is easy to state: Can we find integers x,y,z such that

x³ + y³ + z³ = 42 ?

Professor Andrew Booker (University of Bristol) turned to MIT and collaboration PI Andrew Sutherland to help solve the problem.

Additional links:
MIT News
Numberphile video

Congratulations to Maarten Derickx

Maarten Derickx was awarded the Schoonschip Prize from the Computer Algebra of the Netherlands (CAN) Foundation for best Ph.D. thesis of Dutch universities involving Computer Algebra. Dr. Derickx’s thesis title is Torsion points on elliptic curves on number fields of small degree (Leiden, 2016).

The Schoonschip Prize is named after the first Dutch computer algebra system called Schoonschip. This system was developed in the 1960s by Nobel laureate Martinus J.G. Veltman to assist with renormalizations of gauge theories, and later inspired systems like Mathematica.

Annotating the L-functions and Modular Forms Database

During the week of March 18-22, 2019, a workshop was held at the Institute for Advanced Study (IAS) in Princeton. This workshop had approximately 20 participants, and it was supported in part by the IAS, the American Institute of Mathematics (AIM), and the Simons Foundation. We explored several ways to improve the dissemination of mathematical knowledge via the LMFDB, including: expanding on the current use of “knowls” to display context-free information, implementing annotations which provide context and references associated to individual mathematical objects (as in the OEIS), and implementing a way to visualize the many predictions of the Langlands program within the framework of the LMFDB.

Arnold Ross Lecture on “Elliptic curves”

Collaboration PI Bjorn Poonen will deliver the Arnold Ross Lecture on the topic “Elliptic curves” to an expected audience of over 1000 high school students at the 45th ARML competition at the Pennsylvania State University on May 31, 2019.

Abelian varieties over finite fields

Currently the L-Functions and Modular Forms Database (http://www.lmfdb.org/) includes a database of isogeny classes of abelian varieties over finite fields, identified by their q-Weil polynomials. This meeting set out to enumerate the isomorphism classes within each isogeny class. In the ordinary case, this translates to a problem about complex lattices, and Marseglia has shown how to effectively solve this by enumerating ideal classes in CM orders.

The workshop started by setting up a schema for how we will identify isomorphism classes, including the desired structural data, and we kicked off computations in an initial range. We discussed how to compute polarizations on each isogeny class, and the solution to this problem (enumerating short lattice vectors subject to positivity, up to equivalence) also provides a way to compute isogenies between the isomorphism classes. This will be practical to compute – at least in some range – and we studied examples of isogeny graphs.

A new statistical model challenges long-held assumptions on the possible ranks of elliptic curves.

Quanta Magazine features an article describing recent research involving Simons Collaboration team members
Bjorn Poonen and John Voight, inspired by earlier research by Noam Elkies.
Learn more

Coding sprint at the Arithmetic Geometry, Number Theory, and Computation conference

A coding sprint was part of a follow-up development week for the Arithmetic Geometry, Number Theory, and Computation conference held at MIT from August 20-24. The photo, taken by John Cremona feature “sprinters” Alex Best, David Roe, David Lowry-Douda, Edgar Costa, Maarten Derickx, and John Voight.

Selfridge Prize

Congratulations to Michael Musty, Sam Schiavone, Jeroen Sijsling and John Voight. Their paper “A database of Belyĭ maps” was selected as the winner of the Number Theory Foundation’s 2018 Selfridge Prize in Number Theory, awarded to the best paper submitted to the 13th Algorithmic Number Theory Symposium (ANTS).

Balakrishnan awarded Sloan Fellowship

Collaboration PI Jennifer Balakrishnan has been awarded a Sloan Research Fellowship.

Conference on Arithmetic Geometry, Number Theory, and Computation

The collaboration announced its first conference will be held at MIT this summer, August 20-24, 2018. Learn more.