POTENTIAL AUTOMORPHY AND THE LEOPOLDT CONJECTURE
POTENTIAL AUTOMORPHY AND THE LEOPOLDT CONJECTURE
复制标题
DOI：
10.1353/ajm.2017.0030
复制
发表时间：
2017-10-01
影响因子：
1.7
通讯作者：
Thorne, Jack A.
中科院分区：
文献类型：
--
作者：
Khare, Chandrashekhar B.;Thorne, Jack A.
本文研究了CM域F上GL(N)的Hida的p-进Hecke代数.Hida提出了一个关于这些Hecke代数的维数的猜想，他称之为非交换Leopoldt猜想，并证明了他在F=q的情况下的猜想包含了经典的Leopoldt猜想，如果我们进一步假设扩张K/Q的特征标的自同构诱导的存在，那么他的猜想就是关于Q上n次数域K的经典的Leopoldt猜想.我们用Calegari-Geraghty适应于GLN环境的自同构提升技巧研究了Hida猜想.在这一背景下，我们证明了一个自同构提升结果，条件是对应对称流形的上同调中由扭类产生的Galois表示的存在性和局部-整体相容。在相同的条件下，我们证明了对于全实数域K和素数p的经典(阿贝尔)Leopoldt猜想，可以利用Hida的非阿贝尔Leopoldt猜想推导出Cm域上GL(N)的p-进Hecke代数的经典(阿贝尔)猜想，而不需要假定扩张K/Q的特征标的自同构归纳。
We study in this paper Hida's p-adic Hecke algebra for GL(n) over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case F = Q implies the classical Leopoldt conjecture for a number field K of degree n over Q, if one assumes further the existence of automorphic induction of characters for the extension K/Q. We study Hida's conjecture using the automorphy lifting techniques adapted to the GLn setting by Calegari-Geraghty. We prove an automorphy lifting result in this setting, conditional on existence and local -global compatibility of Galois representations arising from torsion classes in the cohomology of the corresponding symmetric manifolds. Under the same conditions we show that one can deduce the classical (abelian) Leopoldt conjectures for a totally real number field K and a prime p using Hida's non-abelian Leopoldt conjecture for p-adic Hecke algebra for GL(n) over CM fields without needing to assume automorphic induction of characters for the extension K/Q. For this methods of potential automorphy results are used.