2025 Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting
Invitation Only
Meeting Goals:
The last annual meeting of the Simons Collaboration on Arithmetic Geometry, Number Theory & Computation will take stock of achievements over the last seven years and point to promising avenues for future work. Specific topics will include elliptic curves of high rank, rational points on modular curves, new developments in higher-dimensional abelian varieties, modular forms of various flavors, and future growth in the L-functions and Modular Forms Database.
Speakers:
Jennifer Balakrishnan, Boston University
Noam Elkies, Harvard University
Brendan Hassett, Brown University
Bjorn Poonen, MIT
Joseph Silverman, Brown University
John Voight, University of Sydney
• Collaboration Site
• 2020 Annual Meeting
• 2021 Annual Meeting
• 2022 Annual Meeting
• 2023 Annual Meeting
• 2024 Annual Meeting
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Wednesday, January 15, 2025
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Joseph H. Silverman | Height Density and Dynamical Propagation of Rational Points on Varieties 10:30 AM BREAK 11:00 AM Jennifer Balakrishnan | Quadratic Chabauty for Modular Curves 12:00 PM LUNCH 1:00 PM Noam Elkies | Elliptic Curves \(E/{\bf Q}\) of High Rank 2:00 PM BREAK 2:30 PM Lightning Talks
Santiago Arango-Piñeros | Galois Groups of Low-Dimensional Abelian Varieties over Finite Fields
Kate Finnerty | Quadratic Points on Modular Curves in the LMFDB
Sachi Hashimoto | Rational Points on X0(N)* when N is Non-Squarefree
David Lowry-Duda | Rigorous Maass Forms3:30 PM BREAK 4:00 PM Bjorn Poonen | Abelian Varieties and Tetrahedra 5:00 PM DAY ONE CONCLUDES Thursday, January 16, 2025
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Sam Schiavone | Explicit Inverse Galois Theory via Hilbert Modular Forms 10:30 AM BREAK 11:00 AM John Voight | Computing Modular Forms and the LMFDB: Past, Present, and Future 12:00 PM LUNCH 1:00 PM Brendan Hassett | K3 Surfaces, Computation, and Future Challenges 2:00 PM MEETING CONCLUDES -
Jennifer Balakrishnan
Boston UniversityQuadratic Chabauty for Modular Curves
By Faltings’ theorem, the set of rational points on a curve of genus 2 or more is finite. We give a survey of the quadratic Chabauty method, which is used to determine a finite superset of the set of rational points for certain curves of genus 2 or more. In particular, we discuss what aspects of quadratic Chabauty can be made practical for certain modular curves and highlight several examples.
This is based on joint work with Alexander Betts, Netan Dogra, Daniel Hast, Aashraya Jha, Steffen Müller, Jan Tuitman, and Jan Vonk.
Noam Elkies
Harvard UniversityElliptic Curves \(E/{\bf Q}\) of High Rank
Mordell (1922) proved that the rational points of an elliptic curve \(E / {\bf Q}\) form a finitely-generated abelian group. It is still not known which finitely-generated abelian groups can occur as \(E({\bf Q})\). Mazur (1977) proved that the possible torsion subgroups \(T\) are the cyclic groups of order \(1, 2, \ldots, 10\), and \(12\), and the products of cyclic groups of orders \(2\) and \(2k\) with \(k=1,2,3,4\). For each of these fifteen \(T\), it is still an open problem which ranks occur.
For small \(T\), the current records all come from elliptic fibrations of K3 surfaces; the most recent such record is \(29\) for \(|T| = 1\), found in August 2024 and giving the first improvement since 2006 for curves with trivial torsion (and indeed for curves with arbitrary torsion structure). We describe how we find elliptic K3’s over \(\bf Q\) whose Mordell-Weil rank is as high as possible given the torsion subgroup, and how we search for fibers of even higher rank on such a surface.
This joint work with Zev Klagsbrun.
Brendan Hassett
Brown UniversityK3 Surfaces, Computation, and Future Challenges
The last decade has seen breakthroughs in the geometry of K3 surfaces, including results on moduli spaces from the standpoint of Shimura varieties, criteria for good reduction, derived equivalence, and the Kuga-Satake construction. We explore implications of these ideas for computational and effective results.
Bjorn Poonen
MITAbelian Varieties and Tetrahedra
This talk will cover two unrelated topics. The first is constructing abelian varieties over finite fields realizing nearly every possible order in the allowable range (joint work with Raymond van Bommel, Edgar Costa, Wanlin Li, and Alexander Smith). The second is classifying tetrahedra satisfying various geometric conditions, such as tiling space, being scissors-congruent to a cube, or having rational dihedral angles: some of these lead to problems of a familiar type, such as solving a multivariable polynomial equation in roots of unity, but others lead to unsolved unlikely intersection problems in search of a framework
This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein, with further work by MIT undergraduates Abdellatif Anas Chentouf and Yihang Sun.
Joseph H. Silverman
Brown UniversityHeight Density and Dynamical Propagation of Rational Points on Varieties
We start with a 30-year old trichotomy conjecture for the height density of rational points on varieties. We will explain how this conjecture suggests an orbit propagation principle which says that if the rational points are Zariski dense, then the orbits under an endomorphism are widely spaced. As time permits, we will discuss proofs of various versions of the principle for projective spaces, abelian varieties, and surfaces. (Joint work with Hector Pasten)
Sam Schiavone
Massachusetts Institute of TechnologyExplicit Inverse Galois Theory via Hilbert Modular Forms
We construct an explicit polynomial realizing the transitive permutation group 17T7 as a Galois group over the rationals. Our construction uses the 2-torsion of particular abelian fourfolds with real multiplication. We compute such fourfolds using the Eichler-Shimura construction for Hilbert modular forms.
John Voight
University of SydneyComputing Modular Forms and the LMFDB: Past, Present, and Future
Computing modular forms remains a topic of central importance in number theory and arithmetic geometry—both in theory and in practice. In this talk, we survey this topic: after a brief look at the past, we discuss the current state-of-the-art methods and their limitations and then provide some possible future directions. Looking ahead, we consider how collaborative efforts can expand resources like the LMFDB to deepen our explicit understanding of modular forms in the context of the Langlands program.
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Santiago Arango-Piñeros
Emory UniversityGalois Groups of Low-Dimensional Abelian Varieties over Finite Fields
In joint work with Sam Frengley and Sameera Vemulapalli, we consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the \emph{weighted permutation representation} which encompasses all three of these invariants and use it to study the subtle relationships between them. We use this permutation representation to classify the triples of invariants that occur for abelian surfaces and simple abelian threefolds. (https://arxiv.org/abs/2412.03358)
Kate Finnerty
Boston UniversityQuadratic Points on Modular Curves in the LMFDB
We describe a computation of quadratic Chabauty sets of genus 2 bielliptic modular curves over \(\mathbb{Q}\) of ranks 1 and 2 in the LMFDB, building on work of Balakrishnan–Dogra and Bianchi–Padurariu. The analysis produces surprising examples of points over number fields on these curves, including quadratic points on \(X_{ns}^+(15)\). This talk will describe the analysis and briefly describe the results.
Sachi Hashimoto
Brown UniversityRational Points on X0(N)* when N is Non-Squarefree
The rational points of the modular curve \(X_0(N)\) classify pairs \((E,C_N)\) of elliptic curves over \(Q\) together with a rational cyclic subgroup of order \(N\). The curve \(X_0(N)^*\) is the quotient of \(X_0(N)\) by the full group of Atkin-Lehner involutions. Elkies showed that the rational points on this curve classify elliptic curves over the algebraic closure of \(Q\) that are isogenous to their Galois conjugates. In ongoing joint work with Timo Keller and Samuel Le Fourn, we study the rational points on the family \(X_0(N)^*\) for \(N\) non-squarefree. In particular we report on an integrality result for \(X_0(N)^*\).
David Lowry-Duda
Institute of Computational and Experimental Research in MathematicsRigorous Maass Forms
Maass forms are non-holomorphic modular forms that are eigenfunctions of the Laplace Beltrami operator. Despite being building blocks for all classical modular forms, they’re challenging to compute. In this lightning talk, we’ll describe past, present, and future efforts towards collections of rigorous Maass forms.
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Overview
In-person participants will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.
Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting (a meeting held at another institution, for example) will not be reimbursed by the Simons Foundation and should be paid by other sources.
Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.
Meals
The daily meal limit is $125; itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses up the following amounts.
- Breakfast $20
- Lunch $30
- Dinner $75
Allowable Meal Expenses
- Meals taken on travel days (when you traveled by air or train).
- Meals not provided on a meeting day, dinner on Friday for example.
- Group dinners consisting of fellow meeting participants paid by a single person will be reimbursed up to $75 per person and the amount will count towards each individual’s $125 daily meal limit.
Unallowable Meal Expenses
- Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner).
- Meals taken on days not associated with Simons Foundation-coordinated events.
- Minibar expenses.
- Meal expenses for a non-foundation guest.
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Ubers, Lyfts, taxis, etc., taken to and from restaurants in Manhattan.
- Accommodations will be made for those with mobility restrictions.
Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport or train station) as well as subway and bus fares while in Manhattan are reimbursable.
Transportation to/from satellite meetings are not reimbursable.
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Air and Rail
For funded individuals, the foundation will arrange and pay for round-trip travel from their home city to the conference.
All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency.
Travel Deviations
The following travel specifications are considered deviations and will only be accommodated if the cost is less than or equal to the amount the Simons Foundation would pay for a standard round-trip ticket from your home city to the conference city:
- Preferred airline
- Preferred travel class
- Specific flights/flight times
- Travel dates outside those associated with the conference
- Arriving or departing from an airport other than your home city or conference city airports, i.e. multi-segment or triangle trips.
All deviations must be reviewed and approved by the Simons Foundation and, if the cost is in excess of what would normally be paid, a reimbursement quote must be obtained through the foundation’s travel agency before proceeding to booking and paying for travel out of pocket. All reimbursements for travel booked directly will be paid after the conclusion of the meeting.
Changes After Ticketing
All costs related to changes made to ticketed travel are to be paid for by the participant and are not reimbursable. Please contact the foundation’s travel agency for further assistance.
Personal & Rental Cars
Personal car and rental trips over 250 miles each way require prior approval from the Simons Foundation via email.
Rental cars must be pre-approved by the Simons Foundation.
The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.
Hotel
Funded individuals who require hotel accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel, arriving one day before the meeting and departing one day after the meeting.
Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link included in your invitation.
The James NoMad Hotel
22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)
https://www.jameshotels.com/new-york-nomad/For driving directions to The James NoMad, please click here.
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Overview
In-person participants will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.
Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting (a meeting held at another institution, for example) will not be reimbursed by the Simons Foundation and should be paid by other sources.
Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.
Meals
The daily meal limit is $125; itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses up the following amounts.
- Breakfast $20
- Lunch $30
- Dinner $75
Allowable Meal Expenses
- Meals taken on travel days (when you traveled by air or train).
- Meals not provided on a meeting day, dinner on Friday for example.
- Group dinners consisting of fellow meeting participants paid by a single person will be reimbursed up to $75 per person and the amount will count towards each individual’s $125 daily meal limit.
Unallowable Meal Expenses
- Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner).
- Meals taken on days not associated with Simons Foundation-coordinated events.
- Minibar expenses.
- Meal expenses for a non-foundation guest.
-
Ubers, Lyfts, taxis, etc., taken to and from restaurants in Manhattan.
- Accommodations will be made for those with mobility restrictions.
Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport or train station) as well as subway and bus fares while in Manhattan are reimbursable.
Transportation to/from satellite meetings are not reimbursable.
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Attendance
In-person participants and speakers are expected to attend all meeting days. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.
Entry & Building Access
Upon arrival, guests will be required to show their photo ID to enter the Simons Foundation and Flatiron Institute buildings. After checking-in at the meeting reception desk, guests will be able to show their meeting name badge to re-enter the building. If you forget your name badge, you will need to provide your photo ID.
The Simons Foundation and Flatiron Institute buildings are not considered “open campuses” and meeting participants will only have access to the spaces in which the meeting will take place. All other areas are off limits without prior approval.
If you require a private space to conduct a phone call or remote meeting, please contact your meeting manager at least 48-hours ahead of time so that they may book a space for you within the foundation’s room reservation system.
Guests & Children
Meeting participants are required to give 24 hour advance notice of any guests meeting them at the Simons Foundation either before or after the meeting. Outside guests are discouraged from joining meeting activities, including meals.
With the exception of Simons Foundation and Flatiron Institute staff, ad hoc meeting participants who did not receive a meeting invitation directly from the Simons Foundation are not permitted.
Children under the age of 18 are not permitted to attend meetings at the Simons Foundation. Furthermore, the Simons Foundation does not provide childcare facilities or support of any kind. Special accommodations will be made for nursing parents.
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Meeting & Policy Questions
Meghan Fazzi
Senior Manager, Events & Administration, MPS
[email protected]Travel & Hotel Support
FCM Travel Meetings & Events
[email protected]
Hours: M-F, 8:30 AM-5:00 PM ET
+1-877-300-7108