4
votes
1answer
184 views

Iwasawa logarithm and analytic continuation

I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$. ...
5
votes
2answers
231 views

Absolutely irreducible p-adic representation of the absolute Galois group of Q_p

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, $G_p$ the absolute Galois group of $\mathbb{Q}_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}_p$. Assume we are ...
3
votes
2answers
280 views

Trace of n-th root of unity in cyclotomic extension of p-adic rationals

Let $n\in\mathbb N$ and $p$ be any prime. Denote by $\mathbb Q_p$ the $p$-adic numbers. For a field extension $L/K$ denote by $Tr_{L/K}$ the corresponding trace function. Let $\zeta_n$ be a primitve ...
7
votes
2answers
321 views

convergence in Z-hat; modulo prime power

The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16). Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$ by $a_0=b, a_{n+1}=2^{a_n}$. Prove that ...
0
votes
0answers
234 views

What is $p$-adic Fourier series?

Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$? Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion: ...
5
votes
0answers
252 views

maximal abelian extension of quadratic extension of $\mathbb Q_p$

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb ...
5
votes
0answers
200 views

2-adic Logarithm and Resistance of n-dimensional Cube

Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is $$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=1}^{n}\frac1{{n-1\choose k}}.$$ The ...
4
votes
3answers
311 views

structure of norm one group for quadratic extension of p-adic fields

Let $F$ be a p-adic field (finite extensions of $\mathbb{Q}_p$ for some prime $p$), and $E/F$ be a quadratic extension. Use $\sigma$ to denote the nontrivial element in the Galois group $Gal(E/F)$. ...
6
votes
1answer
193 views

$(\varphi, \Gamma)$-modules of finite height

Maybe the answer to my question is obvious. Let $p$ be a prime $\geq 3$. Let $D$ be an ├ętale $(\varphi, \Gamma)$-module over $A_{\mathbb{Q}_p} = \{ \sum_{n \in \mathbb{Z}} a_n X^n \, \vert \, a_n \in ...
6
votes
2answers
452 views

Algebraic $p$-adic integers mod $p$

In studying the papers of Abbes and Saito on ramification theory in the imperfect residue field case, I come to the following questions. Let $\overline{\mathbb{Q}}_p\supset\mathbb{Q}{}^{t}_p\supset ...
2
votes
1answer
222 views

Trivial p-adic measures

I am looking at p-adic distributions, and in this case p-adic measures. To say that $\mu$ is a distribution means that the arguments of $\mu$ are compact open subsets of $\mathbb{Z}_p$, $\mu$ is ...
7
votes
0answers
391 views

The character of a separable degree-$p$ extension of a local field of residual characteristic $p$ ?

Let $p$ be a prime number and $F$ a finite extension of ${\mathbf Q}_p$ or of ${\mathbf F}_p((t))$. I'm going to define a natural map from the set ${\mathcal S}_p(F)$ of degree-$p$ separable ...