数字域上签名$(p,p,\text{3})$的三元丢番图方程

Pub Date : 2022-06-24 DOI:10.4153/S0008414X22000311
Erman Isik, Yasemin Kara, Ekin Ozman
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引用次数: 1

摘要

摘要本文用模方法证明了Diophantine方程$x^p+y^p=z^3$在各种数域上解的结果。首先,通过假设一些标准模性猜想,证明了一类窄类的一般数域满足某些技术条件的渐近结果。其次,我们证明了存在一个显式边界,使得方程$x^p+y^p=z^3$在$K=\mathbb {Q}(\sqrt {d})$上没有特定类型的解,其中$d=1,7,19,43,67$,当p大于这个边界时。在证明过程中,我们证明了伽罗瓦表示、惯性群象和比安奇新形式的不可约性的各种结果。
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On ternary Diophantine equations of signature $(p,p,\text{3})$ over number fields
Abstract 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. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, 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 $K=\mathbb {Q}(\sqrt {-d})$ , 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.
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