AI 中文总结
该研究针对虚二次域，通过定义范退化概型证明了864个p=3、203个p=5、全部3个p=7的相关数域的p类塔群为温和的，是首批被证明此类性质的数域。
AI 中文摘要
设K为虚二次数域，p为奇素数，G_∅(K)(p)为K的极大处处非分歧pro-p扩张的伽罗瓦群。对该群的每个模p特征χ，我们定义二次范算子D_χ：Cl(K)[p]→Cl(K)/p。这些算子决定了H¹(G_∅(K)(p), F_p)上的所有三重Massey乘积，进而决定该群的三次初始关系。在p类秩为3时，映射χ→D_χ的2×2子式定义了F_p上射影平面P²的一个子概型，我们将其称为K的范退化概型。我们证明：若该概型在对角Massey乘积的零点轨迹上存在一个既约孤立闭点，则G_∅(K)(p)是温和的，因此具有上同调维数2。我们处理所有满足|D_K|<2³⁰且奇p类秩为3的虚二次域，涉及的素数仅为p=3、5、7。我们无条件证明：在p=3的12749个域中，有864个域的p类塔群是温和的；在p=5的204个域中，有203个是温和的；在p=7的3个域全部是温和的。据我们所知，这些是首批被证明其极大处处非分歧pro-p伽罗瓦群为温和的数域，尤其提供了明确的无限p类塔，其伽罗瓦群具有上同调维数2。
英文摘要
Let $K$ be an imaginary quadratic number field, let $p$ be an odd prime, and let $G=G_{\varnothing}(K)(p)$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\mathrm{Cl}(K)[p]$ to $\mathrm{Cl}(K)/p$; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-$p$ extension attached to $x$. These maps determine all triple Massey products on $H^1(G,\mathbb F_p)$, and with them the cubic initial relations of $G$. Suppose that the $p$-class rank $d=\dim_{\mathbb F_p}\mathrm{Cl}(K)/p$ is at least three. The $(d-1)\times(d-1)$ minors of the family $x\mapsto D_x$ define a subscheme $Σ_D$ of $\mathbb P^{d-1}_{\mathbb F_p}$, an invariant of $K$, the norm-degeneracy scheme. We prove: if the rank condition $\mathrm{rk}\,D_x=d-2$ holds transversally at a point of $Σ_D$, over some finite extension of $\mathbb F_p$, then $G$ is mild, and hence of cohomological dimension 2. For $p>3$ transversality means that $Σ_D$ is smooth of dimension $d-3$ at the point; at $p=3$ the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of $p$-class rank at least three with $|D_K|<2^{30}$. Only $p=3$, 5, and 7 occur. The criterion decides 206 of the 207 fields at $p=5$ and 7, and 505 of the 12 750 fields at $p=3$, where the Bockstein condition restricts its reach. A direct computation with the cubic initial relations settles the remaining field at $p=5$ and a further 11 765 at $p=3$, 26 of them not mild, while 480 remain undecided. In all, the $p$-class tower group is proved mild for 12 451 of the 12 957 fields. These appear to be the first number fields for which the full maximal everywhere unramified pro-$p$ Galois group is proved to be mild, and hence of cohomological dimension 2.
Comments70 pages; code and data: doi:10.5281/zenodo.22255983. v3: the range |D_K|<2^30 is now fully treated; of 12 957 fields of p-class rank >=3, 12 451 are proved mild, while the 26 not mild and the 480 undecided are all at p=3; the last p=5 field and the rank-four field are settled, and a rank-four census to 2*10^10 is added; abstract and Section 5 reorganised, criterion unchanged
