论文标题
在伪造的数字字段
On fake subfields of number fields
论文作者
论文摘要
我们研究了局部全球原则在“数字园”方面的失败；即，我们对成对的数字字段$（k_1，k_2）$感兴趣，因此$ k_2 $不是$ k_1 $的任何代数的con轭$ k_1^σ$的子场k_1^σ$。这种情况的示例自然出现，但不仅是通过算术上等效的数字字段的概念而出现的。我们提供了一些系统的结构，产生了“假子场”（从上述意义上），这些构造不是由算术等价引起的。这也可以解释为与数字字段的Zeta函数有关的某些局部全球原理的失败。
We investigate the failure of a local-global principle with regard to "containment of number fields"; i.e., we are interested in pairs of number fields $(K_1,K_2)$ such that $K_2$ is not a subfield of any algebraic conjugate $K_1^σ$ of $K_1$, but the splitting type of any single rational prime $p$ unramified in $K_1$ and in $K_2$ is such that it cannot rule out the containment $K_2\subseteq K_1^σ$. Examples of such situations arise naturally, but not exclusively, via the well-studied concept of arithmetically equivalent number fields. We give some systematic constructions yielding "fake subfields" (in the above sense) which are not induced by arithmetic equivalence. This may also be interpreted as a failure of a certain local-global principle related to zeta functions of number fields.
