## Experimental Mathematics

- Experiment. Math.
- Volume 6, Issue 4 (1997), 273-287.

### 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.

#### Article information

**Source**

Experiment. Math., Volume 6, Issue 4 (1997), 273-287.

**Dates**

First available in Project Euclid: 7 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1047047188

**Mathematical Reviews number (MathSciNet)**

MR1606908

**Zentralblatt MATH identifier**

1115.14304

**Subjects**

Primary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

Secondary: 14H40: Jacobians, Prym varieties [See also 32G20]

#### Citation

Weber, Hermann-Josef. Hyperelliptic simple factors of {$J\sb 0(N)$} with dimension at least {$3$}. Experiment. Math. 6 (1997), no. 4, 273--287. https://projecteuclid.org/euclid.em/1047047188