2013
DOI: 10.1007/s12188-013-0080-4
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Modular forms of orthogonal type and Jacobi theta-series

Abstract: In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A 1 = 2 ). We give a lot of examples of Jacobi forms of singular and critical weights for root systems using Jacobi theta-series. We give the Jacobi lifting for Jacobi forms of half-integral indices. In some case it gives additive lifting construction of new reflective modular forms.

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Cited by 37 publications
(77 citation statements)
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“…In Table 1, for the special case of level one, antisymmetric and odd weight are equivalent. The following theorem gives the general result on the parity of D 0 in terms of the hypotheses of Theorem 6.6 and thereby completes the proof of item (3) in that theorem.…”
Section: Proof Of Symmetry and Antisymmetrysupporting
confidence: 53%
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“…In Table 1, for the special case of level one, antisymmetric and odd weight are equivalent. The following theorem gives the general result on the parity of D 0 in terms of the hypotheses of Theorem 6.6 and thereby completes the proof of item (3) in that theorem.…”
Section: Proof Of Symmetry and Antisymmetrysupporting
confidence: 53%
“…As above, according to the divisor principle, we have Borch(ψ 6,3 ) = Grit(φ 6,3 ). The same argument shows that this is the only Siegel cusp form for K(3) of weight 6, a fact first proved in [18].…”
Section: N=1 To Construct Holomorphic Borcherds Products Inmentioning
confidence: 89%
See 1 more Smart Citation
“…The Jacobi forms of half‐integral indices were introduced in . (See for the lattice case.) We define the following Jacobi form of singular weight 4 for D8ϑD8false(τ,frakturzfalse)=ϑfalse(z1false)··ϑfalse(z8false)J4,D8,where z=false(z1,,z8false)D8C where the coordinates zi correspond to the euclidean basis of the model of Dn.…”
Section: Reflective Towers Of Jacobi Liftingsmentioning
confidence: 99%
“…For example, ψ2,4A1false(τ,z4false)=ϑfalse(τ,z1false)ϑfalse(τ,z2false)ϑfalse(τ,z3false)ϑfalse(τ,z4false)J2,D4false(vη12false)is a D4‐Jacobi form with character vη12:SL2false(double-struckZfalse)false{±1false}. The same product ψ2,4A1false(τ,z4false)=ϑfalse(τ,z1false)ϑfalse(τ,z2false)ϑfalse(τ,z3false)ϑfalse(τ,z4false)J2,4A1;12false(vη12×vHfalse)is a Jacobi form of index 12 with respect to the lattice 4A1 where vH is the unique non‐trivial binary character of the Heisenberg group H(4A1) (see ). For such Jacobi forms, the corresponding lifting contains only Hecke operators of indices congruent to a constant modulo the conductor of the character.…”
Section: Reflective Towers Of Jacobi Liftingsmentioning
confidence: 99%