Organizing Committee
Abstract

In this workshop, we will explore a number of themes in the arithmetic of abelian varieties of low dimension (typically dimension 2–4), with a focus on computational aspects. Topics will include the study of torsion points, Galois representations, endomorphism rings, Sato-Tate distributions, Mumford-Tate groups, complex and p-adic analytic aspects, L-functions, rational points, and so on. We also seek to classify and tabulate these objects, in particular to understand explicitly the underlying moduli spaces (with specified polarization, endomorphism, and torsion structure), and to find examples of abelian varieties exhibiting special behavior. Finally, we will pursue connections with related areas, including the theory of modular forms, related algebraic varieties (e.g., K3 surfaces), and special values of L-functions.

Our goal is for the workshop to bring together researchers working on abelian varieties in a number of facets to establish collaborations, develop algorithms, and stimulate further research.

Questions proposed in the workshop problem session


This workshop is an activity of the Simons Collaboration 'Arithmetic Geometry, Number Theory, and Computation' and is supported by the Simons Foundation.

Image for "Arithmetic of Low-Dimensional Abelian Varieties"

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, June 3, 2019
TimeEventLocationMaterials
8:30 - 8:55am EDTRegistration - ICERM 121 South Main Street, Providence RI 0290311th Floor Collaborative Space 
8:55 - 9:00am EDTWelcome - ICERM Director11th Floor Lecture Hall 
9:00 - 9:50am EDTHearing algebraic curves and factoring polynomials - Felipe Voloch, University of Canterbury11th Floor Lecture Hall
10:00 - 10:30am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20am EDTLocal-global principles for endomorphism rings of abelian varieties - Davide Lombardo, Università di Pisa11th Floor Lecture Hall
11:35 - 12:25pm EDTEndomorphisms and decompositions of Jacobians - Jeroen Sijsling, Universität Ulm11th Floor Lecture Hall
12:35 - 2:30pm EDTBreak for Lunch / Free Time  
2:30 - 3:20pm EDTDecomposing Jacobian Varieties - Jen Paulhus, Grinnell College11th Floor Lecture Hall
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDTArithmetic of hyperelliptic curves over local fields - Celine Maistret, Boston University11th Floor Lecture Hall
5:00 - 6:30pm EDTWelcome Reception11th Floor Collaborative Space 
Tuesday, June 4, 2019
TimeEventLocationMaterials
9:00 - 9:50am EDTOn uniformity conjectures for abelian varieties and K3 surfaces over number fields - Yuri Zarhin, Pennsylvania State University11th Floor Lecture Hall
10:00 - 10:30am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20am EDTLow degree points on curves - Isabel Vogt, Stanford University11th Floor Lecture Hall
11:35 - 12:25pm EDTAbelian Divisibility Sequences - Joe Silverman, Brown University11th Floor Lecture Hall
12:35 - 2:30pm EDTBreak for Lunch / Free Time  
2:30 - 3:30pm EDTLightning Talks Session 111th Floor Lecture Hall
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm EDTLightning Talks Session 211th Floor Lecture Hall
Wednesday, June 5, 2019
TimeEventLocationMaterials
9:00 - 9:50am EDTPrym varieties of genus four curves - Nils Bruin, Simon Fraser University11th Floor Lecture Hall
10:00 - 10:30am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20am EDTSato-Tate groups of abelian threefolds - Francesc Fité, IAS11th Floor Lecture Hall
11:35 - 12:25pm EDTProblem Session11th Floor Lecture Hall
12:35 - 12:40pm EDTGroup Photo11th Floor Lecture Hall 
12:40 - 3:30pm EDTBreak for Lunch / Free Time  
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDTFree Time11th Floor Lecture Hall 
Thursday, June 6, 2019
TimeEventLocationMaterials
9:00 - 9:50am EDTCounting isogenous principally-polarized abelian varieties over finite fields - Everett Howe, Center for Communications Research, La Jolla11th Floor Lecture Hall
10:00 - 10:30am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20am EDTArithmetic of twists of supersingular abelian varieties - Valentijn Karemaker, University of Pennsylvania11th Floor Lecture Hall
11:35 - 12:25pm EDTNewton Polygon Stratification of the Torelli Locus in PEL-type Shimura Varieties - Wanlin Li, University of Wisconsin-Madison11th Floor Lecture Hall
12:35 - 2:30pm EDTBreak for Lunch / Free Time  
2:30 - 3:20pm EDTUsing Abelian Varieties for Multiparty non-Interactive Key Exchange - Alice Silverberg, University of California, Irvine11th Floor Lecture Hall
3:30 - 4:00pm EDTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:50pm EDT Advances in Isogeny-based Cryptography - Benjamin Smith, Inria and École Polytechnique11th Floor Lecture Hall
Friday, June 7, 2019
TimeEventLocationMaterials
9:00 - 9:50am EDTClutching of curves, Newton polygons, and unlikely intersections. - Rachel Pries, Colorado State University11th Floor Lecture Hall
10:00 - 10:30am EDTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20am EDTAbelian varieties with everywhere good reduction over certain real quadratic fields of small discriminant - Lassina Dembele, Dartmouth College11th Floor Lecture Hall
11:35 - 12:25pm EDTExceptional Vinberg representations and moduli spaces - Eric Rains, California Institute of Technology11th Floor Lecture Hall
12:35 - 2:30pm EDTBreak for Lunch / Free Time  

Lecture Videos

Exceptional Vinberg representations and moduli spaces

Eric Rains
California Institute of Technology
June 7, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Abelian varieties with everywhere good reduction over certain real quadratic fields of small discriminant

Lassina Dembele
Dartmouth College
June 7, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Clutching of curves, Newton polygons, and unlikely intersections

Rachel Pries
Colorado State University
June 7, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract

Advances in Isogeny-based Cryptography

Benjamin Smith
Inria and École Polytechnique
June 6, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Newton Polygon Stratification of the Torelli Locus in PEL-type Shimura Varieties

Wanlin Li
University of Wisconsin
June 6, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract

Arithmetic of twists of supersingular abelian varieties

Valentijn Karemaker
University of Pennsylvania
June 6, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract

Counting isogenous principally-polarized abelian varieties over finite fields

Everett Howe
Center for Communications Research La Jolla
June 6, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Sato-Tate groups of abelian threefolds

Francesc Fité
IAS
June 5, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Prym varieties of genus four curves

Nils Bruin
Simon Fraser University
June 5, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Abelian Divisibility Sequences

Joseph Silverman
Brown University
June 4, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Low degree points on curves

Isabel Vogt
Stanford University
June 4, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract

On uniformity conjectures for abelian varieties and K3 surfaces over number fields

Yuri Zarhin
Pennsylvania State University
June 4, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Arithmetic of hyperelliptic curves over local fields

Celine Maistret
Boston University
June 3, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract

Decomposing Jacobian Varieties

Jennifer Paulhus
Grinnell College
June 3, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Endomorphisms and decompositions of Jacobians

Jeroen Sijsling
Universität Ulm at the Institut für Reine Mathematik
June 3, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract     Slides

Local-global principles for endomorphism rings of abelian varieties

Davide Lombardo
Università di Pisa
June 3, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract

Hearing algebraic curves and factoring polynomials

Jose Voloch
University of Canterbury
June 3, 2019
Arithmetic of Low-Dimensional Abelian Varieties
Play     Abstract