# Theta Function

## Definition

Let $V$ be a complex vector space and $\Gamma$ a lattice in $V$ (a discrete subgroup of $V$ that spans $V$ as a real vector space).

Definition(theta function) . A theta function $\theta:V\mapsto\mathbb{C}$ is a holomorphic function on $V$ that is quasi-periodic with respect to $\Gamma$. Namely, for each $\gamma\in \Gamma$ there is a holomorphic map $e_\gamma:V\to \mathbb{C}$, s.t. $$\theta(z+\gamma) = e_\gamma(z)\theta(z).$$

## Theta Correspondence

Given a symplectic vector space $(V,\omega)$ we have the corresponding Heisenberg group $\mathsf{H}(V)$. It admits a unique irreducible representation in a Hilbert space in which the center $U(1)$ acts by homotheties. Endow $V$ with a complex structure such that $\omega$ becomes the imaginary part of some Hermitian inner product gives rise to a model, called the Fock space $\mathcal{F}$, for this unique representation. If $\omega$ takes integral values on a lattice $\Gamma\subset V$, then $\Gamma$ lifts as a subgroup of $\mathsf{H}(V)$.

Theta functions appear naturally as $\Gamma$-invariants in $\mathcal{F}$ or an enlarged space $\mathcal{F}_{-\infty}$.