Twists of X(7) and primitive solutions to x^2+y^3=z^7
Twists of X(7) and primitive solutions to x^2+y^3=z^7
复制标题
X(7) 的扭曲和 x^2 y^3=z^7 的原始解
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
M. Stoll
中科院分区：
文献类型：
--
作者：
B. Poonen;Edward F. Schaefer;M. Stoll
我们找到x^2+y^3=z^7的原始整数解。一个nonabelian下降参数涉及简单的168阶群减少了问题的确定合理的一组点上的一组有限的扭曲的克莱因四次曲线X。为了限制相关扭曲的集合，我们利用X和模曲线X（7）之间的同构，并使用椭圆曲线的模性和水平降低。这留下了10个亏格-3曲线，其有理点通过组合方法找到。
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus-3 curves, whose rational points are found by a combination of methods.