Modular Forms and Curves of Low Genus: Computational Aspects
(September 28 - October 2, 2015)

Description

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.

NOTE: Participants of this workshop may also be interested in registering for the fall 2015 Algebraic Geometry Northeastern Series (AGNES) conference being held at Brown University from October 2-October 4, as well as a ICERM's public lecture "Lotteries and Geometries" the evening of Friday, October 2.

Organizing Committee

= speaker   = poster presenter

Monday September 28, 2015
Time Description Speaker Location Abstracts Slides
8:30 - 8:55Registration121 S.Main Street, Providence, RI (11th Floor)
8:55 - 9:00WelcomeICERM Director11th Floor Lecture Hall
9:00 - 9:45Shimura Degrees, New Modular Degrees, and Congruence PrimesAlyson Deines, CCR - La Jolla11th Floor Lecture Hall
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10:00 - 10:45Elliptic Curves with Good Reduction Outside SAngelos Koutsianas, University of Warwick11th Floor Lecture Hall
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10:45 - 11:15Coffee/Tea Break11th Floor Collaborative Space
11:15 - 12:00Obtaining modular units via a recurrence relationMarco Streng, University of Leiden11th Floor Lecture Hall
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12:00 - 2:00Break for Lunch
2:00 - 2:45Shadow lines in the arithmetic of elliptic curvesRachel Newton, Institut des Hautes Études Scientifiques (IHÉS)11th Floor Lecture Hall
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3:00 - 3:45Variations on quadratic ChabautyJennifer Balakrishnan, Oxford University11th Floor Lecture Hall
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3:45 - 4:15Coffee/Tea Break
4:15 - 5:00Non-archimedean construction of elliptic curves and abelian surfacesMarc Masdeu, University of Warwick11th Floor Lecture Hall
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5:00 - 6:30Welcome Reception11th Floor Collaborative Space

Tuesday September 29, 2015
Time Description Speaker Location Abstracts Slides
9:00 - 9:45Constructing genus 2 curves over finite fieldsKirsten Eisentraeger, Pennsylvania State11th Floor Lecture Hall
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10:00 - 10:45Genus two CM curves and their invariantsTonghai Yang, University of Wisconsin11th Floor Lecture Hall
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10:45 - 11:15Coffee/Tea Break11th Floor Collaborative Space
11:15 - 12:00Isogenies, polarisations and real multiplicationDamien Robert, Ecole Normale Superieure11th Floor Lecture Hall
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12:00 - 12:10Group Photo11th Floor Lecture Hall
12:10 - 2:00Break for Lunch
2:00 - 2:45Computing Fourier coefficients of theta seriesReinier Broker, Brown University11th Floor Lecture Hall
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3:00 - 3:45Genus 3 curves and the inverse Galois problemSamuele Anni, University of Warwick11th Floor Lecture Hall
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3:45 - 4:15Coffee/Tea Break
4:15 - 5:00Explicit methods in the theory of Jacobi forms of lattice index and over number fieldsNils Skoruppa, Universitat Siegen11th Floor Lecture Hall
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Wednesday September 30, 2015
Time Description Speaker Location Abstracts Slides
9:00 - 9:45On the paramodular conjectureArmand Brumer, Fordham University11th Floor Lecture Hall
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10:00 - 10:45A survey on the modularity of abelian surfacesLassina Dembele, University of Warwick11th Floor Lecture Hall
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10:45 - 11:15Coffee/Tea Break
11:15 - 12:00On the paramodular conjectureAriel Pacetti, University of Buenos Aires11th Floor Lecture Hall
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12:00 - 1:45Break for Lunch
1:45 - 2:30Tables of paramodular eigenformsCris Poor, Fordham University11th Floor Lecture Hall
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2:45 - 3:30Modularity of the abelian surface of conductor 277David Yuen, Lake Forest College11th Floor Lecture Hall
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3:30 - 5:00Coffee/Tea Break and Poster Session11th Floor Collaborative Space

Thursday October 1, 2015
Time Description Speaker Location Abstracts Slides
11:15 - 12:00Quadratic twists of Siegel modular forms of paramodular level- Hecke operators and Fourier coefficientsJennifer Johnson-Leung, University of Idaho11th Floor Lecture Hall
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12:00 - 2:00Break for Lunch
2:00 - 2:45Constructing genus 3 hyperelliptic Jacobians - a mathematician's point of viewChristelle Vincent, Stanford University11th Floor Lecture Hall
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3:00 - 3:45Constructing genus 3 hyperelliptic Jacobians with CM - a cryptographer's point of viewSorina Ionica, Ecole Normale Superieure11th Floor Lecture Hall
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3:45 - 4:15Coffee/Tea Break
4:15 - 5:00Bad reduction of genus 3 curves with complex multiplicationElisa Lorenzo Garcia, Universiteit Leiden11th Floor Lecture Hall
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Friday October 2, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 10:45Computing the images of Galois representationsDavid Zywina, Cornell University11th Floor Lecture Hall
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10:45 - 11:15Coffee/Tea Break
11:15 - 12:00Computing L-series of hyperelliptic curves in moderate genusDavid Harvey, University of New South Wales11th Floor Lecture Hall
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12:15 - 1:00Torsion subgroups of rational elliptic curves over the compositum of all cubic fieldsAndrew Sutherland, MIT11th Floor Lecture Hall
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1:00 - 3:15Break for Lunch
3:15 - 3:45Coffee/Tea Break