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On iterated extensions of number fields arising from quadratic polynomial maps
Journal of Number Theory ( IF 0.6 ) Pub Date : 2020-04-01 , DOI: 10.1016/j.jnt.2019.08.022
Kota Yamamoto

Abstract A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x 2 − 2 , and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p 2 .

中文翻译:

关于由二次多项式映射产生的数域的迭代扩展

摘要 质数 p 和基点 β 的后临界有限有理映射 ϕ 产生了数域的有限分枝迭代扩展的塔,并且有时提供具有 p-adic Lie 图像的树栖伽罗瓦表示。在本文中,我们将 ϕ 取为 monic Chebyshev 多项式 x 2 − 2 ,并检查结果塔中理想扩展类群的 2 部分的大小。在某些情况下,我们证明了来自岩泽理论的格林伯格猜想的类似物。一个关键的工具是关于 p 2 次相对循环扩展的类数的 p 不可分性的一般定理。
更新日期:2020-04-01
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