Asymptotic lower bound of class numbers along a Galois representation
Asymptotic lower bound of class numbers along a Galois representation
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沿着伽罗瓦表示的类数的渐近下界
DOI：
10.1016/j.jnt.2019.09.024
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发表时间：
2018
影响因子：
0.7
通讯作者：
T. Ohshita
中科院分区：
文献类型：
--
作者：
T. Ohshita
设T是有限秩的自由Zp-模，它具有数域K的绝对Galois群的连续Zp-线性作用，且满足一定条件.利用T对应的一个塞尔默群，给出了由T/pn T的稳定子固定的伽罗瓦扩张域Kn的类数的可加p-adic赋值的一个下界.应用这个结果，我们证明了一个渐近不等式，它描述了在一定条件下，给定的交换簇A在Mordell-Weil群下，类数沿着塔K（A [p∞]）/K的一个显式下界.我们还证明了当A是Hilbert-Blumenthal或CM阿贝尔簇时的另一个渐近不等式。
Let T be a free Z p-module of finite rank equipped with a continuous Z p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of K n, which is the Galois extension field of K fixed by the stabilizer of T/p n T. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K (A [p∞])/K for a given abelian variety A with certain conditions in terms of the Mordell–Weil group. We also prove another asymptotic inequality for the cases when A is a Hilbert–Blumenthal or CM abelian variety.