Non‐commutative Iwasawa theory for modular forms
Non‐commutative Iwasawa theory for modular forms
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DOI：
10.1112/plms/pds061
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发表时间：
2012-03
影响因子：
1.8
通讯作者：
J. Coates;T. Dokchitser;Z. Liang;W. Stein;R. Sujatha
中科院分区：
文献类型：
--
作者：
J. Coates;T. Dokchitser;Z. Liang;W. Stein;R. Sujatha
本文的目的是给出证据，主要是数值证据，以支持岩泽理论的非交换主要猜想，即通过邻接所有p-幂单位根和固定整数m > 1的所有p-幂根获得的Galois扩张上的权k > 2的本原模形式的动机。在这种情况下，主要猜想的预测是相当复杂的，因为有一个以上的临界点，也没有规范的周期选择。然而，我们的数值数据完全符合主要猜想的所有方面，包括加藤在分圆马宁p-adic L-函数和分圆p-adic L-函数之间的神秘同余，该分圆p-adic L-函数的动机被该扩张的伽罗瓦群的某个非阿贝尔阿丁特征标扭曲。
The aim of the present paper is to give evidence, largely numerical, in support of the non‐commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k > 2 over the Galois extension of ℚ obtained by adjoining to ℚ all p‐power roots of unity, and all p‐power roots of a fixed integer m > 1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agree perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p‐adic L‐function, and the cyclotomic p‐adic L‐function of a twist of the motive by a certain non‐abelian Artin character of the Galois group of this extension.