p-rigidity and Iwasawa μ-invariants
p-rigidity and Iwasawa μ-invariants
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p-刚度和 Iwasawa μ-不变量
DOI：
10.2140/ant.2017.11.1921
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发表时间：
2017
影响因子：
1.3
通讯作者：
H. Hida
中科院分区：
文献类型：
--
作者：
Ashay A. Burungale;H. Hida
设F是一个全真实的域，其整数环为O，p是F中的非分歧奇素数.设p是p上的素数，我们证明了与F相关联的mod p Hilbert模形式是由它对部分Serre-Tate变形空间Gm ∈ Op（p-刚性）的限制决定的.设K/F是一个虚二次CM扩张，使得F中p以上的每个素数在K中分裂，λ是K的Hecke特征标.部分基于p-刚性，证明了λ的反分圆Katz p-adic L-函数的μ-不变量等于λ的全反分圆Katz p-adic L-函数的μ-不变量.对于一类Rankin-Selberg p-adic L-函数，也有类似的结果.当λ是根数为−1的自对偶时，我们证明了λ的Katz p-adic L-函数的分圆导数的μ-不变量等于λ的Katz p-adic L-函数的分圆导数的μ-不变量.基于作者和Hsieh以前的工作，我们得到了这些p-adic L-函数和导数的μ-不变量的一个公式，在大多数情况下.我们还证明了Gillard猜想的一个p-版本，即λ的Katz p-adic L-函数的μ-不变量为零.
Let F be a totally real field with ring of integers O and p be an odd prime unramified in F. Let p be a prime above p. We prove that a mod p Hilbert modular form associated to F is determined by its restriction to the partial Serre-Tate deformation space Gm ⊗ Op (p-rigidity). Let K/F be an imaginary quadratic CM extension such that each prime of F above p splits in K and λ a Hecke character of K. Partly based on p-rigidity, we prove that the µ-invariant of anticyclotomic Katz p-adic L-function of λ equals the µ-invariant of the full anticyclotomic Katz p-adic L-function of λ. An analogue holds for a class of Rankin-Selberg p-adic L-functions. When λ is self-dual with the root number −1, we prove that the µ-invariant of the cyclotomic derivatives of Katz p-adic L-function of λ equals the µ-invariant of the cyclotomic derivatives of Katz p-adic L-function of λ. Based on previous works of authors and Hsieh, we consequently obtain a formula for the µ-invariant of these p-adic L-functions and derivatives, in most of the cases. We also prove a p-version of a conjecture of Gillard, namely the vanishing of the µ-invariant of Katz p-adic L-function of λ.