论文标题
签名$（4,2，p）$在数字字段上
Asymptotic Fermat for signature $(4,2,p)$ over number fields
论文作者
论文摘要
令$ k $为一个数字字段。使用模块化方法，我们证明了$ x^4-y^2 = z^p $ over $ k $的解决方案的解决方案的渐近结果，假设$ k $至少具有一个复杂的嵌入时，则假设Langlands计划的一些深层但标准的猜想。另一方面，在完全真正的扩展名的情况下，我们给出了无条件的结果。当已知椭圆曲线的模块化时，例如，当$ k $是真正的Quadratic或$ r $ - $ r $ - layer的$ \ mathbb {z} _2 $ extension $ \ mathbb {q} $的$ extension，有效的渐近结果。
Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^4-y^2=z^p$ over $K$, assuming some deep but standard conjectures of the Langlands programme when $K$ has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above $2$. When modularity of elliptic curves over $K$ is known, for example when $K$ is real quadratic or the $r$-layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, effective asymptotic results hold.
