论文标题
$ \ mathbb {q} $ - 曲线，Hecke字符和一些Diophantine方程
$\mathbb{Q}$-curves, Hecke characters and some Diophantine equations
论文作者
论文摘要
在本文中，我们研究了方程$ x^4+dy^2 = z^p $和$ x^2+dy^6 = z^p $，用于$ d $的正平方值。 $ \ mathbb {q}（\ sqrt {-d}）$上的frey曲线被连接到每个原始解决方案，恰好是$ \ mathbb {q} $ - 曲线。我们的主要结果是构建了Hecke角色$χ$满足，即$χ$扭曲的Frey椭圆曲线表示延伸到$ \ \ text {gal} _ \ Mathbb {q} $，因此（（由Serre的Cuntures）对应于$ s_2（n，\ varepsilon $ new form $ new for newfors $ n $ new form $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $。遵循一些众所周知的结果和消除技术（以及一些改进），它提供了一个系统的程序来研究上述方程式的解决方案，并使我们能够证明不存在针对两个方程式的$ p $ $ p $的非平凡原始解决方案，用于$ d $的新值。
In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $χ$ satisfying that the Frey elliptic curve representation twisted by $χ$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (together with some improvements) it provides a systematic procedure to study solutions of the above equations and allows us to prove non-existence of non-trivial primitive solutions for large values of $p$ of both equations for new values of $d$.
