Finding rational points on bielliptic genus 2 curves
Finding rational points on bielliptic genus 2 curves
复制标题
在双椭圆 2 曲线上寻找有理点
DOI：
10.1007/s002290050215
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发表时间：
1999
影响因子：
0.6
通讯作者：
J. L. Wetherell
中科院分区：
文献类型：
--
作者：
E. V. Flynn;J. L. Wetherell
翻译后摘要：我们讨论了一种技术，试图找到所有合理的点的形式Y2= f3 X6 + f2 X4 + f1 X2 +f0，其中六次曲线具有非零判别式的曲线。这是亏格为2的双椭圆曲线。当雅可比行列式的秩为0或1时，可以应用Chabauty定理。然而，我们将集中讨论秩至少为2的情况。在这种情况下，我们将导出定义在数域<$α上的相关椭圆曲线族。如果这些椭圆曲线中的每一个的秩都小于α的次数：
然后，我们将描述一种类似于Chabauty的技术，该技术可用于试图找到定义在椭圆曲线上的所有点（x，y），其中x∈ φ。这反过来又允许我们找到原始亏格2曲线上的所有双有理点。我们应用这给一个问题的解决丢番图（其中六次曲线在X是不可约的在X上），这简化了最近的解决方案Wetherell。我们还提出了两个例子，其中X中的六次曲线是可约的。
Abstract:We discuss a technique for trying to find all rational points on curves of the form Y2=f3X6+f2X4+f1X2+f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα :
ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ.