Let $p$ be an odd prime. For a number field $K$, we let $K_{\infty }$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\text{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite $p$-groups can be extended in a consistent way to the infinite pro-$p$ groups which can arise in both the real and imaginary quadratic settings.

中文翻译：

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真实二次场的类塔的启发式

让$p$是一个奇数素数。对于数字字段$K$，我们让$K_{\infty}$是最大的无分支亲$p$的扩展$K$; 我们叫这个小组$\text{Gal}(K_{\infty }/K)$这$p$级塔组$K$. 在之前的工作中，作为 Cohen 和 Lenstra 关于理想类群工作的非阿贝尔概括，我们研究了给定有限$p$-组作为$p$一个虚构的二次场的级塔群。在这里，我们对任意实数二次场做同样的事情$K$作为基础。和之前一样，动作$\text{Gal}(K/\mathbb{Q})$在$p$级塔组$K$起着至关重要的作用；然而，地面场中无限阶单元的存在显着地复杂化了可能发生的组的可能性。我们还使用波士顿和伍德的想法，通过删除某个假设来加强我们在虚构二次场案例中的结果。在附录中，我们展示了如何引入有限的概率$p$-组可以以一致的方式扩展到无限亲$p$可以在实数和虚数二次设置中出现的组。