arXiv 1803.040472018-03-11math.NT

实二次域p-类塔的启发式算法

Heuristics for p-class towers of real quadratic fields

Nigel Boston, Michael R. Bush, Farshid Hajir

摘要：

设$p$为奇素数。对于数域$K$，我们令$K_\infty$是$K$的最大非分歧pro-$p$扩张;我们称群$\mathrm{Gal}(K_\infty/K)$为$K$的$p$ -类塔群。在以前的工作，作为一个非阿贝尔推广的科恩和Lenstra的工作，理想类群，我们研究了如何可能是一个给定的有限$p$ -组出现的$p$ -类塔群的虚二次域。这里我们对任意实二次域$K$作同样的处理。如前所述，$\mathrm{Gal}(K/\mathbb{Q})$对$K$的$p$类塔群的作用起着至关重要的作用;然而，在基域中存在无限阶单元，这显著地使可能发生的群的可能性复杂化。我们还锐化我们的结果在虚二次域的情况下，通过删除一定的假设，使用波士顿和伍德的想法。在附录中，我们展示了有限$p$ -群引入的概率如何以一致的方式扩展到无限的pro- $p$群，这些群可以在实二次和虚二次设置中出现。

英文摘要：

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 $\mathrm{Gal}(K_\infty/K)$ 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 $\mathrm{Gal}(K/\mathbb{Q})$ 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 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 an 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.}

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文