Deadline to Register
Friday December 10, 2021
Brendan Hassett, Brown University
The 2022 Simons Collaboration on Arithmetic Geometry, Number Theory and Computation Annual Meeting will focus on the following themes:
- Development and organization of software and databases supporting research in number theory and arithmetic geometry
- Fundamental research in arithmetic geometry inspired by computation and leading to new algorithms
- Explorations of L-functions, modular forms, and Galois representations with elegant and unusual properties
Talks will present contributions from members of the collaboration and work by leading experts that may inspire future developments.
Thursday, January 13
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Brendan Hassett | Rationality and Arithmetic 10:30 AM BREAK 11:00 AM Yunqing Tang | The Unbounded Denominators Conjecture 12:00 PM LUNCH 1:00 PM David Roe | Finite Groups in the LMFDB 2:00 PM BREAK 2:30 PM Lightning Talk Session I 3:30 PM BREAK 4:00 PM Lightning Talk Session II 5:00 PM DAY ONE CONCLUDES
Friday, January 14
8:30 AM CHECK-IN & BREAKFAST 9:30 AM David Zureick-Brown | l-adic Images of Galois for Elliptic Curves Over Q 10:30 AM BREAK 11:00 AM Ila Varma | Geometry of Numbers Methods for Counting in the Cusp 12:00 PM LUNCH 1:00 PM Robert Lemke Oliver | The Structure of Imprimitive Number Fields 2:00 PM MEETING CONCLUDES
Rationality and Arithmetic
Given a family of smooth, projective varieties, which fibers are rational? Hassett will discuss arithmetic, geometric and cycle-theoretic invariants, shedding light on this question. This talk is based on work done in collaboration with Frei, Kollár, Tschinkel and Várilly-Alvarado.
Robert Lemke Oliver
The Structure of Imprimitive Number Fields
Number fields tautologically come in two flavors: those with subfields and those without. Number fields without subfields are called primitive, and those with subfields are called imprimitive. Arithmetic statistics is stuck on many of its fundamental problems for families of primitive fields. Lemke Oliver will report on recent work with collaborators, including Jiuya Wang and Melanie Matchett Wood, on systematic approaches to some of these problems for imprimitive number fields.
Massachusetts Institute of Technology
Finite Groups in the LMFDB
Finite groups arise in many areas of mathematics. Within the L-functions and modular forms database, they show up as Galois groups, as automorphism groups of higher genus curves, and as component groups of Sato-Tate groups. Various databases of finite groups already exist, most notably the small groups database for orders up to 2,000 and the transitive groups database for degree up to 48. Roe will describe efforts in the past several years to incorporate these into the LMFDB, to make them available online and to extend them. In addition to giving a demo of the results, Roe will highlight some interesting mathematics that arose along the way, including finding human-friendly presentations, computing subgroups up to automorphism and finding the minimal faithful degrees of permutation and linear representations of a finite group.
The Unbounded Denominators Conjecture
The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, Tang will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. This talk is based on joint work with Frank Calegari and Vesselin Dimitrov.
University of Toronto
Geometry of Numbers Methods for Counting in the Cusp
In joint work with Arul Shankar, Artane Siad and Ashwin Swaminathan, Varma developed a new method for counting integral orbits having bounded invariants and satisfying congruence conditions that lie inside the cusps of fundamental domains for coregular representations (i.e., representations of semi-simple groups for which the ring of invariants is a polynomial ring). During this talk, Varma will illustrate this method in the case of counting 3-torsion elements in class groups of quadratic orders, and time permitting, he will discuss the new applications of these methods, including to counting 2-torsion ideal classes of monogenized degree-n orders.
l-adic Images of Galois for Elliptic Curves Over Q
Zureick-Brown will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s Program B, the classification of the possible ‘images of Galois’ associated with an elliptic curve (equivalently, classification of all rational points on certain modular curves XH). The main result is a provisional classification of the possible images of l-adic Galois representations associated to elliptic curves over Q and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.
Zureick-Brown will discuss the framework and various applications (e.g., a very fast algorithm to rigorously compute the l-adic image of Galois of an elliptic curve over Q) and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagyn’s theorem to the modular curves we study.
Participation & Funding
Participation in the meeting falls into the following four categories. An individual’s participation category is communicated via their letter of invitation.
Group A – PIs and Speakers
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Business-class or premium economy airfare will be booked for all flights over five hours.
Group B – Out-of-town Participants
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Economy-class airfare will be booked for all flights.
Group C – Local Participants
Individuals in Group C will not receive financial support, but are encouraged to enjoy all conference-hosted meals.
Group D – Remote Participants
Individuals in Group D will participate in the meeting remotely. Please register at the link above and a remote participation link will be sent to you approximately two weeks prior to the meeting.
Travel & Hotel
Air and Train
The foundation will arrange and pay for all air and train travel to the conference for those in Groups A and B. Please provide your travel specifications by clicking the registration link above. If you are unsure of your group, please refer to your invitation sent via email.
For participants in Groups A & B driving to Manhattan, 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.
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.
The James NoMad Hotel
22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)
For driving directions to The James NoMad, please click here.
ALL in-person meeting attendees must be vaccinated against the COVID-19 virus with a World Health Organization approved vaccine, be beyond the 14-day inoculation period of their final dose, and provide proof of vaccination upon arrival to the conference. Acceptable vaccines can be found at the bottom of this page on WHO’s site.
Individuals in Groups A & B will be reimbursed for meals not hosted by the Simons Foundation as well as local expenses, including ground transportation. Additional information in this regard will be emailed on the final day of the meeting.