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Organizing Committee
Wake Forest University
Tufts University
Institute for Advanced Study
Brown University
Amherst College
Abstract
In recent years, there has been an explosion of activity surrounding algebraic points on curves, from many different perspectives. These include the study of measures of irrationality, isolated and parametrized points, computational methods to determine algebraic points, and the arithmetic statistics of algebraic points. In this workshop, we aim to bring together researchers from these diverse perspectives, with the particular goal of developing bridges between them. The workshop will include overview talks on the various perspectives, research talks, an open problem session, and structured time for collaboration.
The deadline to apply for this workshop is February 15, 2025.
The deadline to apply for this workshop is February 15, 2025.
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Workshop Schedule
Monday, June 23, 2025
8:30 - 8:45 AM EDT
Check In
10th Floor Collaborative Space
8:45 - 8:55 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
8:55 - 9:00 AM EDT
Organizer Welcome
Welcome - 11th Floor Lecture Hall
Abbey Bourdon, Wake Forest University
Robert Lemke Oliver, Tufts University
Ari Shnidman, Institute for Advanced Study
Isabel Vogt, Brown University
David Zureick-Brown, Amherst College
9:00 - 10:00 AM EDT
Parameterized and isolated points on curves
11th Floor Lecture Hall
Speaker
Bianca Viray, University of Washington
Session Chair
Isabel Vogt, Brown University
Abstract
The Mordell Conjecture (proved by Faltings in 1983) is a landmark result exemplifying the philosophy "Geometry controls arithmetic". However, it also implies that we can never hope to understand the arithmetic of a higher genus curve solely by studying its rational points over a fixed number field. In this talk, we will introduce the concepts of parametrized points, isolated points, and density degree sets and show how they, together with the Mordell-Lang conjecture (proved by Faltings in 1994), allow us to organize all algebraic points on a curve.
10:15 - 10:45 AM EDT
Coffee Break
11th Floor Collaborative Space
10:45 - 11:45 AM EDT
Sources of low degree points on curves
11th Floor Lecture Hall
Virtual Speaker
Borys Kadets, Hebrew University
Session Chair
Isabel Vogt, Brown University
Abstract
I will describe an approach to understanding low degree points on curves using discrete geometry of subspace configurations, along with applications to points of degrees up to 5 and Hilbert irreducibility theorems for sufficiently ramified coverings. The talk is based on joint work with Isabel Vogt and Danny Neftin.
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 2:30 PM EDT
A database of modular curves
11th Floor Lecture Hall
Speaker
Andrew Sutherland, MIT
Session Chair
David Zureick-Brown, Amherst College
Abstract
The L-functions and Modular Forms Database (LMFDB) was recently expanded to include a database containing tens of millions of modular curves that includes information about local obstructions, genus, gonality, analytic rank, morphisms, and the isogeny decomposition of their Jacobians, which are directly relevant to questions about rational (and low-degree) points on modular curves, as well as links to elliptic curves in the LMFDB that represent non-cuspidal points on these curves. I will give an overview of the construction of this database, the information it currently contains, and the information we would like it to contain in the future (including information about algebraic points).
2:45 - 3:00 PM EDT
Coffee Break
11th Floor Collaborative Space
3:00 - 4:00 PM EDT
Arithmetic Statistics and Algebraic Points
11th Floor Lecture Hall
Speaker
Wei Ho, University of Michigan
Session Chair
David Zureick-Brown, Amherst College
Abstract
We give an overview of methods and ideas from arithmetic statistics that lead to results about rational (and algebraic) points on curves. Much of the heavy lifting is done by Selmer groups.
4:15 - 5:00 PM EDT
Open Problem Session
Problem Session - 11th Floor Lecture Hall
Session Chair
Ari Shnidman, Institute for Advanced Study
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Request Reimbursement
This section is for general purposes only and does not indicate that all attendees receive funding. Please refer to your personalized invitation to review your offer.
- ORCID iD
- As this program is funded by the National Science Foundation (NSF), ICERM is required to collect your ORCID iD if you are receiving funding to attend this program. Be sure to add your ORCID iD to your Cube profile as soon as possible to avoid delaying your reimbursement.
- Acceptable Costs
- 1 roundtrip between your home institute and ICERM
- Flights on U.S. or E.U. airlines – economy class to either Providence airport (PVD) or Boston airport (BOS)
- Ground Transportation to and from airports and ICERM.
- Unacceptable Costs
- Flights on non-U.S. or non-E.U. airlines
- Flights on U.K. airlines
- Seats in economy plus, business class, or first class
- Change ticket fees of any kind
- Multi-use bus passes
- Meals or incidentals
- Advance Approval Required
- Personal car travel to ICERM from outside New England
- Multiple-destination plane ticket; does not include layovers to reach ICERM
- Arriving or departing from ICERM more than a day before or day after the program
- Multiple trips to ICERM
- Rental car to/from ICERM
- Flights on a Swiss, Japanese, or Australian airline
- Arriving or departing from airport other than PVD/BOS or home institution’s local airport
- 2 one-way plane tickets to create a roundtrip (often purchased from Expedia, Orbitz, etc.)
- Travel Maximum Contributions
- New England: $350
- Other contiguous US: $850
- Asia & Oceania: $2,000
- All other locations: $1,500
- Note these rates were updated in Spring 2023 and supersede any prior invitation rates.
- Reimbursement Requests
Request Reimbursement with Cube
Refer to the back of your ID badge for more information. Checklists are available at the front desk and in the Reimbursement section of Cube.
- Reimbursement Tips
- Scanned original receipts are required for all expenses
- Airfare receipt must show full itinerary and payment
- ICERM does not offer per diem or meal reimbursement
- Allowable mileage is reimbursed at prevailing IRS Business Rate and trip documented via Google Maps PDF
- Keep all documentation until you receive your reimbursement!
- Reimbursement Timing
6–8 weeks after all documentation is sent to ICERM. All reimbursements are reviewed by multiple Brown University offices.
- Reimbursement Deadline
Submissions must be received within 30 days of ICERM departure to avoid applicable taxes. No submissions are accepted after six months.