# Theta Functions

One of the main tools for working with analytic Jacobians is the theta function. For instance it is used by [`FromAnalyticJacobian`](../../ArithmeticGeometry/HyperellipticCurves/analytic.md#function-fromanalyticjacobianlabel) and [`RosenhainInvariants`](../../ArithmeticGeometry/HyperellipticCurves/analytic.md#function-rosenhaininvariantslabel). For $c \in {\mathbb{R}}^{2g}$ let $c'$ be the first $g$ entries and $c''$ the second $g$ entries of $c$. For such a $c$, $z \in {\mathbb{C}}^g$ and $\tau$ an element of Siegel upper half-space the classical multi-variable theta function is defined by

$$
\theta[c](z,\tau) = \sum_{m \in {\mathbb{Z}}^g} \exp (\pi i
{}^t(m+c')\tau(m+c') + 2\pi i {}^t(m+c')(z+c'')).
$$

The vector $c$ is called the characteristic of the theta function.

## `Theta(char, z, tau): Mtrx, Mtrx, Mtrx -> FldComElt`

This computes the multidimensional theta function with characteristic $char$ (a $2g \times 1$ matrix) at $z$ (a $g \times 1$ matrix) and $\tau$ (a symmetric $g \times g$ matrix with positive definite imaginary part).

## `Theta(char, z, A): Mtrx, Mtrx, AnHcJac -> FldComElt`

This computes the multidimensional theta function with characteristic $char$ (a $2g \times 1$ matrix) at $z$ (a $g \times 1$ matrix) and $\tau$, the small period matrix of the analytic Jacobian $A$. This function caches the values of theta null values ($z = 0$) at half-integer characteristics.