Organizing Committee
Abstract

One of the crowning achievements of number theory in the 20th century is the construction of the modularity correspondence between elliptic curves with rational coefficients and modular forms of weight 2. The consequences of this result resound throughout number theory; for instance, it enables the resolution of certain problems of diophantine equations (e.g., Fermat's last theorem) as well as the systematic tabulation of elliptic curves, which in turn provides the basis for many new conjectures and results.

The aim of this workshop is to lay the groundwork for extending this correspondence to curves of small genus over number fields. The general framework for this correspondence is predicted by the Langlands program, but much remains to be made explicit. We will explore theoretical, algorithmic, computational, and experimental questions on both sides of the correspondence, with an eye towards tabulation of numerical data and formulation of precise conjectures.

Image for "Modular Forms and Curves of Low Genus: Computational Aspects"

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, September 28, 2015
TimeEventLocationMaterials
8:30 - 8:55am EDTRegistration121 S.Main Street, Providence, RI (11th Floor) 
8:55 - 9:00am EDTWelcome - ICERM Director11th Floor Lecture Hall 
9:00 - 9:45am EDTShimura Degrees, New Modular Degrees, and Congruence Primes - Alyson Deines, CCR - La Jolla11th Floor Lecture Hall
10:00 - 10:45am EDTElliptic Curves with Good Reduction Outside S - Angelos Koutsianas, University of Warwick11th Floor Lecture Hall
10:45 - 11:15am EDTCoffee/Tea Break - 11th Floor Collaborative Space  
11:15 - 12:00pm EDTObtaining modular units via a recurrence relation - Marco Streng, University of Leiden11th Floor Lecture Hall
12:00 - 2:00pm EDTBreak for Lunch  
2:00 - 2:45pm EDTShadow lines in the arithmetic of elliptic curves - Rachel Newton, Institut des Hautes Études Scientifiques (IHÉS)11th Floor Lecture Hall
3:00 - 3:45pm EDTVariations on quadratic Chabauty - Jennifer Balakrishnan, Oxford University11th Floor Lecture Hall
3:45 - 4:15pm EDTCoffee/Tea Break  
4:15 - 5:00pm EDTNon-archimedean construction of elliptic curves and abelian surfaces - Marc Masdeu, University of Warwick11th Floor Lecture Hall
5:00 - 6:30pm EDTWelcome Reception11th Floor Collaborative Space 
Tuesday, September 29, 2015
TimeEventLocationMaterials
9:00 - 9:45am EDTConstructing genus 2 curves over finite fields - Kirsten Eisentraeger, Pennsylvania State11th Floor Lecture Hall
10:00 - 10:45am EDTGenus two CM curves and their invariants - Tonghai Yang, University of Wisconsin11th Floor Lecture Hall
10:45 - 11:15am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:15 - 12:00pm EDTIsogenies, polarisations and real multiplication - Damien Robert, Ecole Normale Superieure11th Floor Lecture Hall
12:00 - 12:10pm EDTGroup Photo11th Floor Lecture Hall 
12:10 - 2:00pm EDTBreak for Lunch  
2:00 - 2:45pm EDTComputing Fourier coefficients of theta series - Reinier Broker, Brown University11th Floor Lecture Hall
3:00 - 3:45pm EDTGenus 3 curves and the inverse Galois problem - Samuele Anni, University of Warwick11th Floor Lecture Hall
3:45 - 4:15pm EDTCoffee/Tea Break  
4:15 - 5:00pm EDTExplicit methods in the theory of Jacobi forms of lattice index and over number fields - Nils Skoruppa, Universitat Siegen11th Floor Lecture Hall
Wednesday, September 30, 2015
TimeEventLocationMaterials
9:00 - 9:45am EDTOn the paramodular conjecture - Armand Brumer, Fordham University11th Floor Lecture Hall
10:00 - 10:45am EDTA survey on the modularity of abelian surfaces - Lassina Dembele, University of Warwick11th Floor Lecture Hall
10:45 - 11:15am EDTCoffee/Tea Break  
11:15 - 12:00pm EDTOn the paramodular conjecture - Ariel Pacetti, University of Buenos Aires11th Floor Lecture Hall
12:00 - 1:45pm EDTBreak for Lunch  
1:45 - 2:30pm EDTTables of paramodular eigenforms - Cris Poor, Fordham University11th Floor Lecture Hall
2:45 - 3:30pm EDTModularity of the abelian surface of conductor 277 - David Yuen, Lake Forest College11th Floor Lecture Hall
3:30 - 5:00pm EDTCoffee/Tea Break and Poster Session11th Floor Collaborative Space 
Thursday, October 1, 2015
TimeEventLocationMaterials
9:00 - 9:45am EDTExperimental results for Siegel modular forms - Nathan Ryan, Bucknell University11th Floor Lecture Hall
10:00 - 10:45am EDTQuadratic twists of Siegel modular forms of paramodular level- motivation and local aspects - Brooks Roberts, University of Idaho11th Floor Lecture Hall
10:45 - 11:15am EDTCoffee/Tea Break11th Floor Collaborative Space 
11:15 - 12:00pm EDTQuadratic twists of Siegel modular forms of paramodular level- Hecke operators and Fourier coefficients - Jennifer Johnson-Leung, University of Idaho11th Floor Lecture Hall
12:00 - 2:00pm EDTBreak for Lunch  
2:00 - 2:45pm EDTConstructing genus 3 hyperelliptic Jacobians - a mathematician's point of view - Christelle Vincent, Stanford University11th Floor Lecture Hall
3:00 - 3:45pm EDTConstructing genus 3 hyperelliptic Jacobians with CM - a cryptographer's point of view - Sorina Ionica, Ecole Normale Superieure11th Floor Lecture Hall
3:45 - 4:15pm EDTCoffee/Tea Break  
4:15 - 5:00pm EDTBad reduction of genus 3 curves with complex multiplication - Elisa Lorenzo Garcia, Universiteit Leiden11th Floor Lecture Hall
Friday, October 2, 2015
TimeEventLocationMaterials
10:00 - 10:45am EDTComputing the images of Galois representations - David Zywina, Cornell University11th Floor Lecture Hall
10:45 - 11:15am EDTCoffee/Tea Break  
11:15 - 12:00pm EDTComputing L-series of hyperelliptic curves in moderate genus - David Harvey, University of New South Wales11th Floor Lecture Hall
12:15 - 1:00pm EDTTorsion subgroups of rational elliptic curves over the compositum of all cubic fields - Andrew Sutherland, MIT11th Floor Lecture Hall
1:00 - 3:15pm EDTBreak for Lunch  
3:15 - 3:45pm EDTCoffee/Tea Break  

Associated Semester Workshops

Computational Aspects of the Langlands Program
Image for "Computational Aspects of the Langlands Program"
Sep 9 - Dec 4, 2015
Semester Program
Computational Aspects of L-functions
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Nov 9 - 13, 2015
Semester Workshop

Lecture Videos

Computing L-series of hyperelliptic curves in moderate genus

David Harvey
University of New South Wales
October 2, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Computing the images of Galois representations

David Zywina
Cornell University
October 2, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Local Galois representations and stable reduction of curves

Irene Bouw
Universität Ulm
October 2, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Constructing genus 3 hyperelliptic Jacobians with CM - a cryptographer's point of view

Sorina Ionica
University of Bordeaux
October 1, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Constructing genus 3 hyperelliptic Jacobians - a mathematician's point of view

Christelle Vincent
University of Vermont
October 1, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Quadratic twists of Siegel modular forms of paramodular level- Hecke operators and Fourier coefficients

Jennifer Johnson-Leung
University of Idaho
October 1, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

Quadratic twists of Siegel modular forms of paramodular level- motivation and local aspects

Brooks Roberts
University of Idaho
October 1, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Experimental results for Siegel modular forms

Nathan Ryan
Bucknell University
October 1, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

Modularity of the abelian surface of conductor 277

David Yuen
Lake Forest College
September 30, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

Tables of paramodular eigenforms

Cris Poor
Fordham University
September 30, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

On the paramodular conjecture

Ariel Pacetti
University of Buenos Aires
September 30, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

A survey on the modularity of abelian surfaces

Lassina Dembele
Warwick University
September 30, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Explicit methods in the theory of Jacobi forms of lattice index and over number fields

Nils Skoruppa
Universität Siegen
September 29, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

Genus 3 curves and the inverse Galois problem

Samuele Anni
University of Warwick
September 29, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Slides

Isogenies, polarisations and real multiplication

Damien Robert
Ecole Normale Superieure
September 29, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

Genus two CM curves and their invariants

Tonghai Yang
University of Wisconsin
September 29, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Constructing genus 2 curves over finite fields

Kirsten Eisentraeger
Pennsylvania State University
September 29, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Non-archimedean construction of elliptic curves and abelian surfaces

Marc Masdeu
University of Warwick
September 28, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides

Variations on quadratic Chabauty

Jennifer Balakrishnan
University of Oxford
September 28, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Obtaining modular units via a recurrence relation

Marco Streng
Universiteit Leiden
September 28, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Elliptic Curves with Good Reduction Outside S

Angelos Koutsianas
Ulm University
September 28, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract

Shimura Degrees, New Modular Degrees, and Congruence Primes

Alyson Deines
Center for Communications Research
September 28, 2015
Modular Forms and Curves of Low Genus: Computational Aspects
Play     Abstract     Slides