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Siegel theta series

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In mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice.

Definition

Suppose that L is a positive definite lattice. The Siegel theta series of degree g is defined by

Θ L g ( T ) = ∑ λ ∈ L g exp ⁡ ( π i T r ( λ T λ t ) ) {\displaystyle \Theta _{L}^{g}(T)=\sum _{\lambda \in L^{g}}\exp(\pi iTr(\lambda T\lambda ^{t}))}

where T is an element of the Siegel upper half plane of degree g.

This is a Siegel modular form of degree d and weight dim(L)/2 for some subgroup of the Siegel modular group. If the lattice L is even and unimodular then this is a Siegel modular form for the full Siegel modular group.

When the degree is 1 this is just the usual theta function of a lattice.

References

  • Freitag, E. (1983), Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften, vol. 254. Springer-Verlag, Berlin, ISBN 3-540-11661-3, MR 0871067{{citation}}: CS1 maint: location missing publisher (link)
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