论文标题
在签名$（p，p，3）$ thimumen字段上的三元diophantine方程
On Ternary Diophantine Equations of Signature $(p,p,3)$ over Number Fields
论文作者
论文摘要
在本文中，我们证明了使用模块化方法在各个数字字段上二世方程的解决方案$ x^p+y^p = z^3 $。首先，通过假设某些标准模块化猜想，我们证明了狭窄班级的一般数字字段的渐近结果满足某些技术条件。其次，我们表明存在明确的界限，以使方程$ x^p+y^p = z^3 $在$ k = \ q（\ sqrt {-d}）上没有特定类型的解决方案，其中$ d = 1,7,19,43,67 $，每当$ p $都大于此限制时，$ d = 1,7,19,43,67 $。在证明过程中，我们证明了有关Galois表示的不可约性，惯性组和Bianchi Newforms的不可约性的各种结果。
In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. Firstly, by assuming some standard modularity conjecture we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Secondly, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\Q(\sqrt{-d})$ where $d=1,7,19,43,67$ whenever $p$ is bigger than this bound. During the course of the proof we prove various results about the irreducibility of Galois representations, image of inertia groups and Bianchi newforms.
