All Questions
9 questions
15
votes
6
answers
1k
views
Conjugacy for $p$-adic matrices of finite order
$\DeclareMathOperator\GL{GL}$Say $p$ is an odd prime, and take two matrices $A,B\in \GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that they are conjugate in $\GL_n({\mathbb Z}_p)$ if and only ...
- 5,363
12
votes
4
answers
688
views
Conjugacy for p-adic matrices of finite order II
Question: Say $p$ is an odd prime, and take two matrices $A,B\in GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that if their reductions mod $p$ are conjugate in $GL_n({\mathbb F}_p)$ then they ...
- 5,363
4
votes
0
answers
211
views
Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
3
votes
0
answers
213
views
Galois action on local deformation ring
Let ${\Bbb Q}_p$ be a local field. For a prime $q \not= p$, we consider an irreducible residual Galois representation
$\overline{\rho} \colon {\mathrm{Gal}}(\overline{{\Bbb Q}}_p/{{\Bbb Q}_p}) \to \...
- 781
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
1
vote
0
answers
174
views
What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
- 7,746
1
vote
0
answers
169
views
Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
0
votes
2
answers
299
views
0-dimensional Gorenstein local ring.
Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring.
Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a ...
- 87
0
votes
1
answer
455
views
Iwasawa theory for Mazur's deformation ring R
The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...
- 87