Is there any literature on multivariable theta functions?

Question

The theta function of a lattice is defined to be $$ \vartheta_\Lambda = \sum_{v\in\Lambda} q^{{\Vert v\Vert}^2}$$ which yields as a coefficient of q^k the number of vectors of norm-squared k.

On the other hand, the Jacobi theta function is given by $$ \vartheta(u,q) = \sum_{n=-\infty}^\infty u^{2n}q^{n^2}$$ and we have the obvious fact that if $\Lambda = \mathbb{Z}$ with its usual intersection form, then $\vartheta(1,q)$ is the theta function for that lattice.

We also have the fact that $\vartheta_{\Lambda_1 \oplus \Lambda_2} = \vartheta_{\Lambda_1}\vartheta_{\Lambda_2}$, and so we can decompose our theta functions into products of theta functions of primitive lattices.

Combining these facts, it is not entirely ridiculous to hope that there is some way to write, for a lattice of rank k, a ``theta function'' of the form $$\vartheta(u_1, \ldots, u_k, q)$$ such that $\vartheta(1, \ldots, 1, q)$ is the ordinary theta function of the lattice. In some sense, the u-variables keep track of the basis elements of the lattice which immediately raises the question as to well-definedness of such an idea; it is worth noting that for the lattice $\Lambda = \bigoplus_i\mathbb{Z}$ that this definition does make sense.

So is there any literature on such objects? Do they make sense for lattices which are not just sums of copies of ℤ? Do they have nice relations akin to those of normal theta functions?

Answer 1

There are three ways to view theta functions

1. as classical homomorphic functions in vector z and/or period matrix T
2. as matrix coefficients of a representation of the Heisenberg and/or Metaplectic grp
3. as sections of Line bundles on the Abelian variety and/or moduli space of the abelian variety

Ram Murty's Theta functions - from the classical to the modern discusses Weil's representation-theoretic interpretation of theta functions. See chapter 3 by Hoffstein on Eisenstein series and theta functions on the metaplectic group. It is the connection to the metaplectic group which gives rise to the functional equation of the multivariable theta function, which you will also find in the chapter on the Metaplectic group in vol 3 of Mumford's Tata Lectures

Bellman's Brief introduction to Theta functions Section 61 alludes to theta functions in several complex variables.

You may also want to search for material in books on Abelian varieties. For example, Baker's Abelian functions Chapter X develops the theory based on the period matrix. Also Murty's book on Abelian varieties and Polishchuk's Abelian Varieties, Theta Functions and the Fourier Transform

Tyurin's Quantization, Classical and Quantum Field Theory, and Theta Functions might also be a useful reference, which I haven't browsed.

See also: Springer Encyclopedia of Math entry on theta functions

Answer 2

I'm not entirely sure what you're seeking, but Wikipedia has a short paragraph on Riemann theta functions. They seem to be a suitable higher-dimensional generalization of the 2-variable Jacobi form you described.

I'm not an expert in this subject, but my understanding is that a theta function is a section of a line bundle on the total space of a universal family of abelian varieties. That is, we have a smooth map $\pi: X_g \to A_g$, where $A_g$ is a moduli space of abelian varieties of dimension $g$ (maybe with a level structure and a polarization), and the fiber over a point on the parameter space describes the abelian variety being parametrized. I think Mumford's Tata Lectures on Theta describe this viewpoint in some detail.