论文标题
在统一根部的存在下，非亚伯群岛 - 朗斯语启发式方法
Non-abelian Cohen--Lenstra Heuristics in the presence of roots of unity
论文作者
论文摘要
对于GALOIS $ k/\ mathbb {f} _q（t）$ galois组$γ$的$ k/\ mathbb {f} _q（T）艾伦伯格（Ellenberg） - 韦卡茨（Venkatesh） - 韦斯特兰（Westerland）和伍德（Wood）引入的Hurwitz计划的不变式。通过跟踪$ω_k$的图像，我们计算为$ k $各种，$ q \ to \ infty $，对于任何$γ$ -group $ h $，其订单为$ q $ q |γ|γ| galois galois的最大不合理扩展名的平均溢出数。在此结果的激励下，我们修改了木材，Zureick-Brown和作者的猜想，内容涉及非亚洲cohen-lenstra，以涵盖基本场包含额外的团结根源时，以涵盖案例。我们还讨论了如何使用不变的$ω_k$来构建随机组模型，并在特殊情况下证明该模型产生与我们功能场结果相同的力矩。
For a Galois extension $K/\mathbb{F}_q(t)$ of Galois group $Γ$ with $\gcd(q,|Γ|)=1$, we define an invariant $ω_K$, and show that it determines the Weil pairing of the curve corresponding to $K$ and it descends to the prime-to-$|Γ|$-torsion part of the lifting invariants of Hurwitz schemes introduced by Ellenberg--Venkatesh--Westerland and Wood. By keeping track of the image of $ω_K$, we compute, as $K$ varies and $q\to \infty$, the average number of surjections from the Galois group of maximal unramified extension of $K$ to $H$, for any $Γ$-group $H$ whose order is prime to $q|Γ|$. Motivated by this result, we modify the conjecture of Wood, Zureick-Brown and the author about non-abelian Cohen--Lenstra, for both function fields and number fields, to cover the cases when the base field contains extra roots of unity. We also discuss how to use the invariant $ω_K$ to construct a random group model, and prove in a special case that the model produces the same moments as our function field result.
