# §21.1 Special Notation

(For other notation see Notation for the Special Functions.)

$g,h$ positive integers. $\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z}$ ($g$ times). $\mathbb{R}\times\mathbb{R}\times\cdots\times\mathbb{R}$ ($g$ times). set of all $g\times h$ matrices with integer elements. $g\times g$ complex, symmetric matrix with $\Im{\boldsymbol{{\Omega}}}$ strictly positive definite, i.e., a Riemann matrix. $g$-dimensional vectors, with all elements in $[0,1)$, unless stated otherwise. $j$th element of vector $\mathbf{a}$. $(j,k)$th element of matrix $\mathbf{A}$. scalar product of the vectors $\mathbf{a}$ and $\mathbf{b}$. $[\boldsymbol{{\Omega}}\mathbf{a}]\cdot\mathbf{b}=[\boldsymbol{{\Omega}}\mathbf% {b}]\cdot\mathbf{a}$. Transpose of $[A_{11},A_{22},\dots,A_{gg}]$. $g\times g$ zero matrix. $g\times g$ identity matrix. $\begin{bmatrix}\boldsymbol{{0}}_{g}&\mathbf{I}_{g}\\ -\mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$. set of $g$-dimensional vectors with elements in $S$. number of elements of the set $S$. set of all elements of the form “$\mbox{element of S_{1}}\times\mbox{element of S_{2}}$”. set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$. (For an example see §20.12(ii).) intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$. line integral of the differential $\omega$ over the cycle $a$.

Lowercase boldface letters or numbers are $g$-dimensional real or complex vectors, either row or column depending on the context. Uppercase boldface letters are $g\times g$ real or complex matrices.

The main functions treated in this chapter are the Riemann theta functions $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, and the Riemann theta functions with characteristics $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$.

The function $\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\mathop{\theta\/}\nolimits\!\left(% \boldsymbol{{\phi}}/(2\pi i)\middle|\mathbf{B}/(2\pi i)\right)$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).