Siegel theta series

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

${\mathrm{\Theta }}_L^g(T)=\sum _{\lambda \in L^g}\exp (\pi iTr(\lambda T{\lambda }^t))$

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

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