Abstract
AbstractWe investigate the behaviour of orthogonal non-holomorphic Eisenstein series at their harmonic points by using theta lifts. In the case of singular weight, we show that the orthogonal non-holomorphic Eisenstein series that can be written as a theta lift have a simple pole at$$s = 1$$s=1whose residues yield holomorphic orthogonal modular forms that are Eisenstein series on the boundary. Moreover, we will investigate the image of this construction and give sufficient conditions for the surjectivity.
Highlights
Let L be an even lattice in a rational quadratic space V of signature (2, l)
As in the higher weight case, it is natural to ask whether all holomorphic orthogonal modular forms of singular weight that are linear combinations of Eisenstein series on the boundary can be obtained in this way
Do induction over the maximal length of chains of cyclic isotropic subgroups containing δ. This means in particular that for F ∈ Mπκ ( (L)) there is a theta lift v for some v ∈ Iso(C[L /L]) whose holomorphic boundary part is given by the boundary part of F (observe that for κ odd the values in 0-dimensional cusps corresponding to δ ∈ Iso(L /L) with δ = −δ must be zero)
Summary
Let L be an even lattice in a rational quadratic space V of signature (2, l). There is an index 2 subgroup of the corresponding orthogonal group O+(2, l) ⊆ O(2, l) acting on the orthogonal upper half-plane Hl. Nz where c0,β (0, 0, s) is a Fourier coefficients of the vector-valued non-holomorphic Eisenstein series E0,β (τ, s) It is a holomorphic orthogonal modular form of singular weight that is an Eisenstein series on the boundary. This is just the Borcherds lift of the invariant vector ress=1 E0,β (τ, s) so that the theorem reads ress=1 0,β (Z , s) = (Z , ress=1 E0,β (·, s)). As in the higher weight case, it is natural to ask whether all holomorphic orthogonal modular forms of singular weight that are linear combinations of Eisenstein series on the boundary can be obtained in this way. If L does not split two hyperbolic planes over Z, we can still fully determine the image of the theta lift (see Corollary 9.7)
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