Deciding Existence of Rational Points on Curves: An Experiment
Deciding Existence of Rational Points on Curves: An Experiment
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确定曲线上有理点的存在：一个实验
DOI：
10.1080/10586458.2008.10129031
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发表时间：
2006
影响因子：
0.5
通讯作者：
M. Stoll
中科院分区：
文献类型：
--
作者：
Nils Bruin;M. Stoll
在本文中，我们收集了与决定曲线是否有有理点的问题相关的实验证据。我们考虑 ℚ 上的所有 genus-2 曲线，由方程 y 2 = f(x) 给出，其中 f 是 5 次或 6 次无平方多项式，积分系数的绝对值至多为 3。对于这大约 200 000 个同构类曲线中的每一个，我们通过组合适用于超椭圆曲线的技术来确定曲线上是否存在有理点。为了开展我们的项目，我们改进和优化了其中一些技术。对于其中 42 条曲线，我们的结果以 Birch 和 Swinnerton-Dyer 猜想或广义黎曼假设为条件。
In this paper we gather experimental evidence related to the question of deciding whether a curve has a rational point. We consider all genus-2 curves over ℚ given by an equation y 2 = f(x) with f a square-free polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200 000 isomorphism classes of curves, we decide whether there is a rational point on the curve by a combination of techniques that are applicable to hyperelliptic curves in general. In order to carry out our project, we have improved and optimized some of these techniques. For 42 of the curves, our result is conditional on the Birch and Swinnerton-Dyer conjecture or on the generalized Riemann hypothesis.