# Theta Function Associated to Kummer Lattice

This is something which I feel must be out in the literature somewhere, but I have been unable to find anything.

If we let $$\text{Km}(A)$$ be the Kummer $$K3$$ surface associated to an abelian surface $$A$$, then we define the Kummer lattice $$K$$ to be the minimal primitive sublattice of $$H^{2}(\text{Km}(A), \mathbb{Z})$$ containing the 16 exceptional curves. The Kummer lattice is negative-definite, even of rank 16 and determinant $$2^{6}$$. For some details, see page 4 of (https://arxiv.org/pdf/1305.3514.pdf).

The theta function associated to $$K$$ is of course

$$\theta_{K}(q, \vec{w}) = \sum_{v \in K} q^{v \cdot v} \, \vec{w}^{v}.$$

Has $$\theta_{K}(q, \vec{w})$$ been studied or described anywhere in the literature? For the life of me, I haven't been able to find a single mention, despite this seeming like a natural question. My guess is that $$\theta_{K}(q, \vec{w})$$ is some sort of lattice Jacobi form of weight $$\frac{16}{2} =8$$. It would be great if this were spelled out or computed somewhere.

For what it's worth, I'm actually interested in the generalized theta function with characteristic

$$\theta_{K}^{(a)}(q, \vec{w}) = \sum_{v \in K} q^{(v+\frac{a}{2}) \cdot (v+\frac{a}{2})} \, \vec{w}^{v+ \frac{a}{2}}$$

for some fixed element $$a \in K$$. If by chance there was also something out there about this more general case that would be great.