Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting 2022

Date & Time


Location

Gerald D. Fischbach Auditorium
160 5th Ave
New York, NY 10010 United States

View Map

Thurs.: 8:30 AM—5 PM
Fri.: 8:30 AM—2 PM

Deadline to Register

Friday December 10, 2021

Invitation Only

Participation is by invitation only. All participants must register.

Organizer:
Brendan Hassett, Brown University

Meeting Goals:

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.

Collaboration Site
2020 Annual Meeting
2021 Annual Meeting

  • Agendaplus--large

    Thursday, January 13

    8:30 AMCHECK-IN & BREAKFAST
    9:30 AM Brendan Hassett | Rationality and Arithmetic
    10:30 AMBREAK
    11:00 AMYunqing Tang | The Unbounded Denominators Conjecture
    12:00 PMLUNCH
    1:00 PMDavid Roe | Finite Groups in the LMFDB
    2:00 PMBREAK
    2:30 PMLightning Talk Session I
    3:30 PMBREAK
    4:00 PMLightning Talk Session II
    5:00 PMDAY ONE CONCLUDES

    Friday, January 14

    8:30 AMCHECK-IN & BREAKFAST
    9:30 AMDavid Zureick-Brown | l-adic Images of Galois for Elliptic Curves Over Q
    10:30 AMBREAK
    11:00 AMIla Varma | Geometry of Numbers Methods for Counting in the Cusp
    12:00 PMLUNCH
    1:00 PMRobert Lemke Oliver | The Structure of Imprimitive Number Fields
    2:00 PMMEETING CONCLUDES
  • Abstractsplus--large

    Brendan Hassett
    Brown University

    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
    Tufts University

    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.
     

    David Roe
    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.
     

    Yunqing Tang
    Princeton University

    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.
     

    Ila Varma
    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.
     

    David Zureick-Brown
    Emory University

    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 & Fundingplus--large

    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 & Hotelplus--large

    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.

    Personal Car
    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.

    Hotel
    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)
    https://www.jameshotels.com/new-york-nomad/

    For driving directions to The James NoMad, please click here.

  • COVID-19 Policyplus--large

    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.

  • Reimbursementplus--large

    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.

  • Contactsplus--large

    Travel & Hotel Assistance
    Ovation Travel Team
    sfnevents@ovationtravel.com
    www.ovationtravel.com

    Meeting Questions & Assistance
    Meghan Fazzi
    Manager, Events and Administration, MPS, Simons Foundation
    mfazzi@simonsfoundation.org
    (212) 524-6080

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