The ants mascot

Thirteenth Algorithmic Number Theory Symposium ANTS-XIII
University of Wisconsin, Madison
July 16 – 20, 2018

Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII)
July 16 – 20, 2018

Analytic evaluation of Hecke eigenvalues for Siegel modular forms of degree two

Owen Colman, Alexandru Ghitza and Nathan C. Ryan

Abstract: The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of F at explicit points in the upper-half space and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is that it returns floating point numbers for the eigenvalues; however, the working precision can be adjusted at will to yield as close an approximation as needed.

Files available: paper (PDF)     slides

© 2017-2018 Jennifer Paulhus (with thanks to Kiran S. Kedlaya, and by extension Pierrick Gaudry and Emmanuel Thomé)