Hyperelliptic Simple Factors of J0(N) with Dimension at Least 3
Hyperelliptic Simple Factors of J0(N) with Dimension at Least 3
复制标题
J0(N) 维数至少为 3 的超椭圆简因式
DOI：
10.1080/10586458.1997.10504615
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发表时间：
1997
影响因子：
0.5
通讯作者：
H. Weber
中科院分区：
文献类型：
--
作者：
H. Weber
我们开发三个问题的算法。从一个g ≥ 2维的复环面开始，同构于一个主极化的简单交换簇A/C，第一个问题是找到超椭圆肖特基问题的算法解：是否存在亏格为g的超椭圆曲线C，其雅可比簇Ic同构于C上的A？我们的解决方案基于[Poor 1994]。如果这样的超椭圆曲线C存在，下一个问题是Rosenhain模型C的构造：y2 = X（X−1）（X−λ1）（X−λ2）...（X−λ2g−1）对于两两不同的数λj E ∈ C \ {0，1}。应用超椭圆theta函数的理论，我们证明了这些数λj可以很容易地用具有偶特征的theta常数来计算。如果阿贝尔簇A被定义在一个域k上（这个域可以是有理数域、低次代数数域或有限域），为了简单起见，我们只在k = Q的情况下证明[Mestre 1991]中的方法是如何推广到得到一个极小方程的。
We develop algorithms for three problems. Starting with a complex torus of dimension g ≥ 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 hvperelliptic curve C of genus g whose jacobian variety Ic 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 C : y2 = X(X−1)(X−λ1)(X−λ2)…(X−λ2g−1) for pairwise distinct numbers λj E ∈ C \ {0, 1}. Applying the theory of hyperelliptic theta functions we show that these numbers λj 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 o...