论文标题
Massey产品在数字字段的典范共同体中
Massey products in the étale cohomology of number fields
论文作者
论文摘要
我们在一个数字字段的整数典范中给出了3倍Massey产品的配方，并使用它们来找到第一个已知的想象中的二次次数领域的示例，其中具有$ p $ - 乘坐的$ p $ -rank两个拥有无限$ p $ p $ p $ p $ -class field Tower，其中$ p $是一个奇怪的prime。此外，就$ p $延伸的班级而言，为消失了3倍的梅西产品而言，这是一种必要和充分的条件。结果，我们为假想的二次场的阶级田地塔的无限层提供了基本和充分的条件。我们还反驳了麦克勒曼的$（3,3）$ - 猜想。最后，我们将Massey产品的消失与$ g _ {\ Mathbb {q}的Galois表示的存在联系起来，S} $实现了一个出乎意料的大型集团，用于二次假想数字字段的某些扩展。
We give formulas for 3-fold Massey products in the étale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an infinite $p$-class field tower, where $p$ is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of $p$-extensions, for the vanishing of 3-fold Massey products is given. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman's $(3,3)$-conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of $G_{\mathbb{Q},S}$ which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field.
