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

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

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)