Surjectivity of Siegel \Phi -operator for square free level and small weight

Computations of Spaces of Paramodular Forms of General Level

“…We briefly mention Satake compactifications and cusps as they pertain to this article, compare [2,35]. For Γ ⊆ Sp 2 (Q) commensurable with Sp 2 (Z), let S(Γ\H 2 ) be the Satake compactification of Γ\H 2 .…”

“…Given such a u, the key entries (3,2) and (4,2) of wutw −1 are multiples of q r . We just need to show there exists e, f, g such that the the remaining key entry (1,2) above is a multiple of q r . Just take e ∈ Z such that (b − aeM − af M 2 zq µ ) is a multiple of q µ+ν ; this is possible because aM is relatively prime to q.…”

“…Given such a u, the expression (8) is already a multiple of q r . Thus it suffices to show there exist e, f, g such that the remaining key entries (1,2) and (3,2) above are multiples of q r . We set g = 0 and leave e, f variable.…”

The Cusp Structure of the Paramodular Groups for Degree Two

“…In [18], Satake proved that for even weights k > 4 the condition (1) characterizes the image of the globalΦ map. The restriction on the weight in his proof arises from the need to have convergent Poincare series; for some relaxation to k = 4 in degree two see [1].…”

“…For even k > 4, by Satake's theorem [18], the codimension of the cusp forms is the dimension of the modular forms in ⊕ m|N M k (Γ m ) that satisfy condition (1) of section 3. All the cusp forms ⊕ m|N S k (Γ m ) satisfy condition (1). The dimension of the Eisenstein series satisfying condition (1) is the number of zero-cusps of S (K(N )\H 2 ) because, in the elliptic modular case for even k ≥ 4, there is a basis of Eisenstein series supported at single cusps, compare the dimension formulae in [4], pages 87-88.…”