Separators of Siegel modular forms of degree two

Abstract

Abstract. We prove that cuspidal Siegel modular forms of degree two and weight 2k are uniquely determined by their Fourier coefficients on small subsets of matrices of content one. This generalizes results of Breulmann, Kohnen, Katsurada, Scharlau and Walling. We give applications to the space of SaitoKurokawa lifts.

Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

“…This result has recently been generalized by Yamana [24] to cusp forms with level and of higher degree. Related results have been obtained by Breulmann-Kohnen [3], Heim [8], Katsurada [9] and Scharlau-Walling [18].…”

Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients

“…Similar results along this line, essentially distinguishing Siegel Hecke eigenforms of degree 2 by the so-called 'radial' Fourier coefficients (i.e., by certain subset of matrices of the form mT with T half-integral, m ≥ 1), has been obtained in Breulmann-Kohnen [2], Scharlau-Walling [18], Katsurada [10]. A result of B. Heim [8] improves upon some of these results using differential operators on Siegel modular forms of degree 2. More recently in [15], [16] A. Saha and R. Schmidt have proved that the Siegel cusp forms of degree 2 are determined (in a quantitative way) by their fundamental (in fact by odd and square-free) Fourier coefficients.…”

Determination of holomorphic modular forms by primitive Fourier coefficients

“…Remark 1 Heim [3] has recently obtained a somewhat elaborate result concerning S κ (Sp 2 (Z)) with even weight κ by essentially the same way.…”