Mild p-Class Tower Groups of Imaginary Quadratic Fields

Abstract

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 $\Sigma_D$ 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 $\Sigma_D$, over some finite extension of $\mathbb F_p$, then $G$ is mild, and hence of cohomological dimension 2. For $p>3$ transversality means that $\Sigma_D$ 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.