虚二次域的$p$类塔的启发式算法，附乔纳森·布莱克赫斯特的附录
Heuristics for $p$-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst
摘要：
Cohen和Lenstra给出了一个启发式的，对于一个固定的奇素数$p$，导致许多有趣的预测分布的$p$类群的虚二次域。我们通过考虑对于每个虚二次域$K$，$K$的$p$类塔的伽罗瓦群，即$G_K:=\mathrm{Gal}(K_\infty/K)$，将Cohen-Lenstra启发式扩展到非阿贝尔设置，其中$K_\infty$是$K$的最大未分歧的$p$ -扩展。利用类域理论，证明了$G_K$的极大交换商同构于$K$的$p$ -类群.对于整数$c\geq 1$，我们给出了一个Cohen-Lentra型的启发式，求出了$\G_K$的最大$p$类$c$商，从而给出了一个表达式，计算出了一个给定的$p$类$c$的$p$ -群以这种方式出现的频率。特别地，我们预测，每一个有限舒尔$σ$ -群发生$G_K$无限多个字段$K$。我们目前的数值数据支持这些acquitures。
英文摘要：
Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\mathrm{Gal}(K_\infty/K)$ where $K_\infty$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\geq 1$, we give a heuristic of Cohen-Lentra type for the maximal $p$-class $c$ quotient of $\G_K$ and thereby give a conjectural formula for how frequently a given $p$-group of $p$-class $c$ occurs in this manner. In particular, we predict that every finite Schur $σ$-group occurs as $G_K$ for infinitely many fields $K$. We present numerical data in support of these conjectures.
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