Computing a Selmer group of a Jacobian using functions on the curve
Computing a Selmer group of a Jacobian using functions on the curve
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使用曲线上的函数计算雅可比行列式的 Selmer 群
DOI：
10.1007/s002080050156
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发表时间：
1998
影响因子：
1.4
通讯作者：
Edward F. Schaefer
中科院分区：
文献类型：
--
作者：
Edward F. Schaefer
一般来说，计算曲线的雅可比矩阵的Selmer群的算法依赖于曲线上的齐次空间或函数。我们对利用曲线上的函数的算法进行了理论分析，并展示了如何利用曲线的特殊性质来生成新的Selmer群计算算法。这种算法的成功将基于我们讨论的两个标准。为了说明可以利用的性质类型，我们开发了一种$(1-\zeta_{p})$- selmer群计算算法，用于形式为$y^{p}=f(x)$的曲线的雅可比矩阵，其中$p$是不除$f$次的素数。我们计算了这种形式的三条曲线的雅可比矩阵的莫德尔-韦尔秩。我们还利用光滑平面四次曲线的双邻线计算了该曲线的雅可比矩阵的2-Selmer群，并用它计算了一个Mordell-Weil秩。
In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show how to exploit special properties of curves to generate new Selmer group computation algorithms. The success of such an algorithm will be based on two criteria that we discuss. To illustrate the types of properties which can be exploited, we develop a $(1-\zeta_{p})$-Selmer group computation algorithm for the Jacobian of a curve of the form $y^{p}=f(x)$ where $p$ is a prime not dividing the degree of $f$. We compute Mordell-Weil ranks of the Jacobians of three curves of this form. We also compute a 2-Selmer group for the Jacobian of a smooth plane quartic curve using bitangents of that curve, and use it to compute a Mordell-Weil rank.