Hyperelliptic simple factors of {$J\sb 0(N)$} with dimension at least {$3$}

Abstract

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.