AI 中文总结
本文从算术代数几何视角推广保持自同构的族的概念，证明一大类profinit群的相关同构族源自内自同构，并将其应用得到ℓ-拟数域的广义Neukirch-Uchida定理。
AI 中文摘要
保持 profinite 群自同构的族是指保持每个 pro-循环子群共轭类的自同构。这一概念用于验证如下性质：某类 profinite 群（如非阿贝尔自由 profinite 群或p进局部域的绝对伽罗瓦群）的每个正规自同构（即保持每个正规闭子群的自同构）都是内自同构，该性质已由Jarden及Jarden-Ritter证明。本文从算术代数几何视角重新考察保持自同构的族这一概念，针对 profinite 群的一般闭子群间的同构，引入该概念的自然推广，将其称为 profinite 群中保持同构的族。对于该推广，本文证明一大类 profinite 群满足如下性质：这类 profinite 群的某些闭子群的保持同构的族源自其自身的内自同构，这类群包括正剩余特征的亨泽尔离散赋值域或Hilbertian域的绝对伽罗瓦群。此外，结合对ℚ的无限代数扩域的绝对伽罗瓦群的pro-循环子群的精细考虑，将该性质作为应用，得到ℓ-拟数域的广义Neukirch-Uchida定理，其中ℓ-拟数域定义为ℚ的代数扩域K，其在ℚ上的伽罗瓦闭包L满足[L:ℚ]不被ℓ^∞整除。令人惊讶的是，该结果包含了任意数域的极大pro-无ℓ扩域的子域类的Neukirch-Uchida型结果。
英文摘要
Families preserving automorphisms of profinite groups are automorphisms that preserve each of the conjugacy classes of pro-cyclic subgroups. This notion appears in the context of verification of the property that every normal automorphism [i.e., an automorphism that preserves each of the normal closed subgroups] of a certain profinite group such as a nonabelian free profinite group or the absolute Galois group of a $p$-adic local field is an inner automorphism, which was proved by Jarden/Jarden-Ritter. In the present paper, we revisit the notion of a families preserving automorphism from the viewpoint of anabelian geometry. We introduce a natural generalized version of this notion for isomorphisms between general closed subgroups of a profinite group, which we shall refer to as families preserving isomorphisms in the profinite group. With regard to this generalized version, we prove that a large class of profinite groups satisfies a property that families preserving isomorphisms of certain closed subgroups of the profinite groups arise from inner automorphisms of them. For instance, the class includes the absolute Galois groups of Henselian discrete valuation fields of positive residue characteristic or Hilbertian fields. Moreover, as an application of this property, together with delicate considerations on pro-cyclic subgroups of the absolute Galois groups of infinite algebraic extension fields of $\mathbb{Q}$, we obtain a generalized version of the Neukirch-Uchida theorem for $l$-quasi-number fields, where an $l$-quasi-number field is defined to be an algebraic extension field $K$ of $\mathbb{Q}$ whose Galois closure $L$ over $\mathbb{Q}$ satisfies that $[L : \mathbb{Q}]$ is not divisible by $l^{\infty}$. Surprisingly, this result includes the Neukirch-Uchida-type result for the class of subfields of the maximal pro-prime-to-$l$ extension fields of arbitrary number fields.
