AI 中文总结
在两个标准数论猜想假设下，利用模方法研究数域上广义费马方程的渐近行为，建立基于$S_K'$-单位方程解的渐近判别准则，针对一类虚二次域证明方程无渐近解，且该类无平方因子整数相对密度为5/6。
AI 中文摘要
设$K$为数域，其整数环为$\u27e8\ua749_K\u27e9$，令$A,B,C \u2208 \u27e8\ua749_K \u27e9\setminus\{0\}$。记$S_K'$为$\u27e8\ua749_K\u27e9$中整除$2ABC$的素理想集合。在关于模$p$伽罗瓦表示的模性以及数域上的Eichler-Shimura（艾希勒-志村）对应的两个标准猜想假设下，我们研究数域$K$上广义费马方程$Ax^p+By^p+Cz^p=0$的渐近行为。利用模方法，我们建立了以关联的$S_K'$-单位方程的解来表述的渐近判别准则。作为应用，我们得到了某些虚二次域$K=\u2102(\u221ad)$上的渐近结果。特别地，对于一族无平方因子整数$d$，我们显式确定了相关的$S_K'$-单位解，并推导出该广义费马方程没有渐近解。最后，我们证明这族无平方因子整数在全体无平方因子正整数中的相对密度为$5/6$。
英文摘要
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspondence over number fields, we study the asymptotic behavior of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$. Using the modular method, we establish an asymptotic criterion in terms of the solutions of the associated $S_K'$-unit equation. As an application, we obtain asymptotic results for certain imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$. In particular, for a family of squarefree integers $d$, we determine the relevant $S_K'$-unit solutions explicitly and deduce that the generalized Fermat equation has no asymptotic solutions. Finally, we show that this family of squarefree integers has relative density $5/6$ among all squarefree positive integers.
Comments15 pages. Comments and suggestions are welcome
