论文标题
Abelian团体可在$ p $ airpary封闭的字段中定义
Abelian groups definable in $p$-adically closed fields
论文作者
论文摘要
回想一下，如果$ G $上有全球类型的$ p $ p $ the $ g $和小型$ m_0 $，则一组$ g $具有有限的令人满意的仿制药（$ fsg $）或可确定的$ f $ generics（$ dfg $）。我们表明，在$ p $ afiper封闭的字段中可以定义的任何亚伯群都是由$ dfg $可确定的组的明确紧凑$ fsg $定义组的扩展。我们讨论一种可能证明可解释的阿伯利亚群体类似陈述的方法。如果$ g $是标准型号$ \ mathbb {q} _p $中可以定义的阿贝里安集团的情况，我们表明$ g^0 = g^{00} $，而$ g $是代数集团的一个开放子组，直到有限因素。后一个结果可以看作是$ \ mathbb {q} _p $中的Abelian可定义组的粗略分类。
Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$.
