论文标题
索引稳定的紧凑型$ p $ -ADIC分析小组
Index-stable compact $p$-adic analytic groups
论文作者
论文摘要
如果任何两个同构开放子组具有相同的索引，则索引组稳定。令$ p $为素数，让$ g $为一个紧凑型$ p $ -Adic Analytic Grout，其中$ \ mathbb {q} _p $ -lie algebra $ \ mathcal {l}（l}（g）$。我们证明，只要$ \ nathcal {l}（g）$是semisimple，$ g $是索引稳定的。特别是，当且仅当它实际上不是Abelian时，仅当时即将到来的紧凑型$ P $ -ADIC分析组就可以稳定了。在紧凑型$ p $ -ADIC分析小组的类别中，这对C. Reid的问题给出了积极的答案。 在附录中，j-p。 Serre证明$ G $在且仅当$ \ Mathcal {l}（g）$具有$ P $ -ADIC NORM 1的任何自动形态的决定因素时才是索引stable。
A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let $p$ be a prime, and let $G$ be a compact $p$-adic analytic group with associated $\mathbb{Q}_p$-Lie algebra $\mathcal{L}(G)$. We prove that $G$ is index-stable whenever $\mathcal{L}(G)$ is semisimple. In particular, a just-infinite compact $p$-adic analytic group is index-stable if and only if it is not virtually abelian. Within the category of compact $p$-adic analytic groups, this gives a positive answer to a question of C. Reid. In the Appendix, J-P. Serre proves that $G$ is index-stable if and only if the determinant of any automorphism of $\mathcal{L}(G)$ has $p$-adic norm 1.
