论文标题
在统一根部存在的情况下
Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity
论文作者
论文摘要
令$ l/k $为全球字段的二次扩展。我们研究了相对类$ g_ {l/k}的$ \ ell $ - 部分的Cohen-lenstra启发式方法：= \ textrm {cl}（l/k）$时，$ k $包含$ \ ell^n $ th的根源。虽然先前已经描述了猜想分布的力矩，但尚无计算给定矩的分布的方法。我们通过引入与类组相关的新不变性，$ψ_{l/k} $和$ω__{l/k}，$解决此问题，并研究$（g_ {l/k}，ψ_{l/k}，ψ_{l/k}的分布，使用线性随机矩阵模型。使用此线性模型，我们在功能场情况下计算分布（包括我们的新不变性），然后在$ \ ell $和$ \ elly $和$ \ infty $上进行局部调整，以在数字字段案例中猜想，这与一些数值实验一致。
Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $\ell$-part of the relative class group $G_{L/K} := \textrm{Cl}(L/K)$ when $K$ contains $\ell^n$th roots of unity. While the moments of a conjectural distribution in this case had previously been described, no method to calculate the distribution given the moments was known. We resolve this issue by introducing new invariants associated to the class group, $ψ_{L/K}$ and $ω_{L/K},$ and study the distribution of $(G_{L/K}, ψ_{L/K}, ω_{L/K})$ using a linear random matrix model. Using this linear model, we calculate the distribution (including our new invariants) in the function field case, and then make local adjustments at the primes lying over $\ell$ and $\infty$ to make a conjecture in the number field case, which agrees with some numerical experiments.
