Working Groups
Group heads are in italics.
The Selberg class: Conrey, Odgers, Farmer, Dehaye, Broughan.
Analytic Tools: Rubinstein, Watkins, Dokchitser, Sound, Montgomery, Booker, Ghaith, Kumar, Broughan, Omar.
Special values of L-functions: Craig, Watkins, Stein, Hanke, Dokchitser, Kedlaya, Harvey, Bradshaw.
Classical modular forms: Stein, Harvey, Bradshaw, Kohel, Oyono, Skoruppa, Walker, Ryan, Farmer, Dembélé.
Maass forms: Farmer, Booker, Stroemberg, Rubinstein.
Higher modular forms: Ryan, Bump, Skoruppa, Dembélé, Kumar, Gunnells, Kohnen, Walling, Caulk, Ghitza.
Design and database issues: Dembélé, Bump, Stein, Kedlaya, Rubinstein, Skoruppa, Kohel, Broughan, Dokchitser, Farmer, Elkies.
Multiple Dirichlet series: Bump, Gunnells, Montgomery, Beineke.
p-adic modular forms and L-values: Harvey, Citro, Kedlaya, Bradshaw, Stein, Burhanuddin.
theta series, lattices, quadratic forms: Voight, Walker, Nebe, Kumar, Hanke, Walling, Kohel
Artin L-functions: Broughan, Dokchitser, Booker, Omar
Projects Proposed Before Workshop
- Create a database of Hilbert modular forms over real quadratic fields.
Produce some numerical evidence for the Langlands correspondence for GSp_4.
- Create a database of Hilbert-Siegel modular forms over real quadratic fields.
- Generating data concerning lifts of Siegel modular forms in positive characteristic to characteristic 0.
- Algorithms for higher rank modular symbols.
- Computing Hecke information for Siegel modular forms.
Developing techniques to compute with p-adic L-functions.
Turning results of the computation of p-adic height pairings on elliptic curves into a database.
Now that Serre's conjecture is a theorem of Khare, Wintenberger, Kisin, et al, the following question seems to be in line. For q a prime power and S a finite set of primes, there are finitely many odd continuous irreducible G_{\mathbf{Q}} \to \mathrm{GL}_2(\mathbf{F}_q) which are unramified outside S; is it practical to list all of them?
- Making an already computed large database of quaternion ideals and associated data for Hecke operators and decompositions.
- Calculating and tabulating data on incomplete Gamma functions.
- Computing data given by weight enumerators of codes and making it into a database.
Accumulating data from the computation of new modular Jacobians of dimension 3 (and new modular curves of genus 3) that are dominated by X_1(N).
Tables of periods of elliptic modular forms of even weight on \Gamma_0(N).
- Tables of Jacobi forms.
- Tables of Siegel modular forms of genus 2 on the full modular group.
- Effective computations of holomorphic modular forms of large weight (at least 10000 or so).
- Computation of Maass waveforms (in various settings) together with their corresponding L-functions.
- To use computing to (1) apply explicit formulas for the action of Hecke operators on Fourier coefficients of a Siegel modular form, and (2) generate data on the Fourier coefficients of Siegel theta series .
- Tables of Heegner points.
- Tables of special values of symmetric power L-functions of elliptic curves.
Projects Proposed During Workshop
- TBA