We develop algorithms for three problems. Starting with a complex
torus of dimension $g\ge 2$, isomorphic to a principally polarized,
simple abelian variety $A$/$\C$, the first problem is to find an
algorithmic solution of the hyperelliptic Schottky problem: Is there a
hyperelliptic curve C of genus g whose jacobian variety
$\mathcal{J}_C$ is isomorphic to A over $\C$? Our solution is based
on [Poor 1994]. If such a hyperelliptic curve C exists, the next
problem is the construction of the Rosenhain model $\medmuskip1mu
C : Y^2=X\mskip1mu
(X-1)(X-\lambda_1)(X-\lambda_2)\,$\dots$(X-\lambda_{2g-1})$ for pairwise
distinct numbers $\lambda_j \in \C\setminus\{0$, $1\}$. Applying the
theory of hyperelliptic theta functions we show that these numbers
$\lambda_j$ can easily be computed by using theta constants with even
characteristics. If the abelian variety A is defined over a field
k (this field could be the field of rational numbers, an algebraic
number field of low degree, or a finite field), we show only in the
case $k=\Q$ for simplicity, how the method in [Mestre 1991] can be
generalized to get a minimal equation over
$\Z\!\left[\frac{1}{2}\right]$ for the hyperelliptic curve C with
jacobian variety $\mathcal{J}_C \cong_{\C} A$. This is our third
problem. For some hyperelliptic, principally polarized and simple
factors with dimension $g=3$, 4, 5 of the jacobian variety
$J_0(N)=\mathcal{J}_{X_0(N)}$ of the modular curve $X_0(N)$ we compute
the corresponding curve equations by applying our algorithms to this
special situation.