Stabile Modulformen und Eisensteinreihen

Siegel Modular Forms and Their Applications

“…Weissauer improved the above result and proved that Φ is surjective if k > (g + r + 3)/2, see [106]. He also treated the case of vector-valued modular forms and showed that the image Φ(M ρ (Γ g )) contains the space of cusp forms…”

On holomorphic projection for symplectic groups

“…With respect to B, we get the dual basis (E ± ) * kl = 1 1+δ kl (E ∓ ) kl as well as B * kl = B lk for all k, l. Proposition 1.1. [13], [24] In terms of the basis above,…”

“…We need some knowledge of the spectral decomposition and extract this out of Langlands' theory of Eisenstein series [10]. (See also [8], [24].) As the action of the center z C of U(g C ) commutes with that of G and U(g C ), it acts by scalars on irreducible components.…”

“…The uniform convergence on F follows from |p τ (z, s)| ≤ det(y) − κ 2 c(s) , and the fact that F is contained in a stripe det y > c > 0. The holomorphicity is equivalent to the formal holomorphicity of the Poincaé series on group level (see [24], p. 49). So p τ is a holomorphic modular form of weight κ ≥ 2m which is square integrable.…”

Class numbers, Jacobi forms and Siegel-Eisenstein series of weight 2 on Sp2 (ℤ)

“…'=Spz(~g) of integral symplectic matrices of size 4. It is known that Ec22)(Z; s) can be continued as a meromorphic function in s to the whole s-plane and is finite at s=0 ([8;12,Theorem 10.4] and [13,Satz 17]). …”