School of Mathematics and Statistics - Research Publications

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Differential operators on modular forms mod p

Ghitza, A (RIMS, 2019)

We give a survey of recent work on the construction of differential operators on various types of modular forms (mod p) . We also discuss a framework for determining the effect of such operators on the mod p Galois representations attached to Hecke eigenforms.
Analytic evaluation of Hecke eigenvalues for Siegel modular forms of degree two

Ghitza, A ; Colman, O ; Ryan, NC (Mathematical Sciences Publishers, 2019)

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.