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.

Confirmed Speakers & Participants

Workshop Schedule

Monday, September 28, 2015
TimeEventLocationMaterials
8:30 - 8:55Registration121 S.Main Street, Providence, RI (11th Floor) 
8:55 - 9:00Welcome - ICERM Director11th Floor Lecture Hall 
9:00 - 9:45Shimura Degrees, New Modular Degrees, and Congruence Primes - Alyson Deines, CCR - La Jolla11th Floor Lecture Hall
10:00 - 10:45Elliptic Curves with Good Reduction Outside S - Angelos Koutsianas, University of Warwick11th Floor Lecture Hall
10:45 - 11:15Coffee/Tea Break - 11th Floor Collaborative Space  
11:15 - 12:00Obtaining modular units via a recurrence relation - Marco Streng, University of Leiden11th Floor Lecture Hall
12:00 - 2:00Break for Lunch  
2:00 - 2:45Shadow lines in the arithmetic of elliptic curves - Rachel Newton, Institut des Hautes Études Scientifiques (IHÉS)11th Floor Lecture Hall
3:00 - 3:45Variations on quadratic Chabauty - Jennifer Balakrishnan, Oxford University11th Floor Lecture Hall
3:45 - 4:15Coffee/Tea Break  
4:15 - 5:00Non-archimedean construction of elliptic curves and abelian surfaces - Marc Masdeu, University of Warwick11th Floor Lecture Hall
5:00 - 6:30Welcome Reception11th Floor Collaborative Space 
Tuesday, September 29, 2015
TimeEventLocationMaterials
9:00 - 9:45Constructing genus 2 curves over finite fields - Kirsten Eisentraeger, Pennsylvania State11th Floor Lecture Hall
10:00 - 10:45Genus two CM curves and their invariants - Tonghai Yang, University of Wisconsin11th Floor Lecture Hall
10:45 - 11:15Coffee/Tea Break11th Floor Collaborative Space 
11:15 - 12:00Isogenies, polarisations and real multiplication - Damien Robert, Ecole Normale Superieure11th Floor Lecture Hall
12:00 - 12:10Group Photo11th Floor Lecture Hall 
12:10 - 2:00Break for Lunch  
2:00 - 2:45Computing Fourier coefficients of theta series - Reinier Broker, Brown University11th Floor Lecture Hall
3:00 - 3:45Genus 3 curves and the inverse Galois problem - Samuele Anni, University of Warwick11th Floor Lecture Hall
3:45 - 4:15Coffee/Tea Break  
4:15 - 5:00Explicit 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 - 9:45On the paramodular conjecture - Armand Brumer, Fordham University11th Floor Lecture Hall
10:00 - 10:45A survey on the modularity of abelian surfaces - Lassina Dembele, University of Warwick11th Floor Lecture Hall
10:45 - 11:15Coffee/Tea Break  
11:15 - 12:00On the paramodular conjecture - Ariel Pacetti, University of Buenos Aires11th Floor Lecture Hall
12:00 - 1:45Break for Lunch  
1:45 - 2:30Tables of paramodular eigenforms - Cris Poor, Fordham University11th Floor Lecture Hall
2:45 - 3:30Modularity of the abelian surface of conductor 277 - David Yuen, Lake Forest College11th Floor Lecture Hall
3:30 - 5:00Coffee/Tea Break and Poster Session11th Floor Collaborative Space 
Thursday, October 1, 2015
TimeEventLocationMaterials
9:00 - 9:45Experimental results for Siegel modular forms - Nathan Ryan, Bucknell University11th Floor Lecture Hall
10:00 - 10:45Quadratic twists of Siegel modular forms of paramodular level- motivation and local aspects - Brooks Roberts, University of Idaho11th Floor Lecture Hall
10:45 - 11:15Coffee/Tea Break11th Floor Collaborative Space 
11:15 - 12:00Quadratic twists of Siegel modular forms of paramodular level- Hecke operators and Fourier coefficients - Jennifer Johnson-Leung, University of Idaho11th Floor Lecture Hall
12:00 - 2:00Break for Lunch  
2:00 - 2:45Constructing genus 3 hyperelliptic Jacobians - a mathematician's point of view - Christelle Vincent, Stanford University11th Floor Lecture Hall
3:00 - 3:45Constructing genus 3 hyperelliptic Jacobians with CM - a cryptographer's point of view - Sorina Ionica, Ecole Normale Superieure11th Floor Lecture Hall
3:45 - 4:15Coffee/Tea Break  
4:15 - 5:00Bad reduction of genus 3 curves with complex multiplication - Elisa Lorenzo Garcia, Universiteit Leiden11th Floor Lecture Hall
Friday, October 2, 2015
TimeEventLocationMaterials
10:00 - 10:45Computing the images of Galois representations - David Zywina, Cornell University11th Floor Lecture Hall
10:45 - 11:15Coffee/Tea Break  
11:15 - 12:00Computing L-series of hyperelliptic curves in moderate genus - David Harvey, University of New South Wales11th Floor Lecture Hall
12:15 - 1:00Torsion subgroups of rational elliptic curves over the compositum of all cubic fields - Andrew Sutherland, MIT11th Floor Lecture Hall
1:00 - 3:15Break for Lunch  
3:15 - 3:45Coffee/Tea Break  

Associated Semester Workshops

Computational Aspects of the Langlands Program
Computational Aspects of L-functions

Lecture Videos