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3
votes
2answers
210 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 ...
6
votes
2answers
309 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
221 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: ...
1
vote
1answer
110 views

a question about a result in Bushnell-Henniart book 'the local Langlands conjecture for GL(2)'

This might be a easy question, but I couldn't get the point. Let $F$ be a p-adic field, $\bar{F}$ a separable algebraic closure of $F$. Set $\Omega_F=Gal(\bar{F}/F)$. Use $F_{\infty}\subset \bar{F}$ ...
5
votes
0answers
241 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
186 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
272 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
153 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 ...
7
votes
0answers
308 views

Torelli-like theorem for K3 surfaces on terms of its étale cohomology

Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology? For example: If $K\ne \mathbb{C} $ and $X\rightarrow ...
45
votes
5answers
2k views

If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds. Monsky's famous and amazingly tricky proof says that if we ...
0
votes
0answers
131 views

shortest relation between poly-Bernoulli numbers and Euler numbers

Poly-Bernoulli numbers which introduced by M.Kaneko are $B_k^{(n)}$ which satisfies in generating function ${Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}$ where Li ...
4
votes
3answers
190 views

Reference for Ostrowski's 1916 Theorem?

I am looking for the original reference for Ostrowski's theorem of 1916 that the only valuations on the rational numbers are the trivial, Archimedean, and p-adic valuations. ...
6
votes
2answers
423 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
207 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 ...
1
vote
0answers
151 views

p-adic Lie theory

It is well known that exponential map in $C^{n\times n}$ will cover all non-sigular matrix $GL(n,C)$, which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases. ...
6
votes
0answers
225 views

Differences in tree picture of ${\bf Q}_p$, $\overline{{\bf Q}_p}$, ${\bf C}_p$, $\Omega_p$

I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value ...
0
votes
0answers
106 views

the definition of pro-infinitesimal thickenings

Let $R$ be a ring and $A$ and $R$-algebra. A pro-infinitesimal thickening of $R$ is a pair $(D, \theta)$ such that $\theta: D \rightarrow R$ is surjective and $D$ is separated and complete for the ...
5
votes
1answer
306 views

Why is this a lattice?

Why is $\text{SL}_3(\mathbf{Z}[1/2])$ a lattice in $\text{SL}_3(\mathbf{R})\times\text{SL}_3(\mathbf{Q}_2)$? Discreteness is pretty clear, but why finite covolume? I understand why ...
2
votes
1answer
327 views

Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.

If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when ...
4
votes
0answers
423 views

Newton Method in $p$-adic case

The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentialbe assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic ...
6
votes
1answer
537 views

Ring of Witt vectors and p-adics

This is probably an easy question, but I'm not able to figure it out. Are the following the same: Field of fractions of the ring of Witt vectors over the algebraic closure of $\mathbb{F}_p$ ...
1
vote
1answer
670 views

Etale cohomology in the $p$-adic setting

Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$? Recall that semialgebraic subsets are obtained from $p$-adic ...
1
vote
1answer
230 views

Partitioning a compact open set into balls in an ultrametric space

Consider a $p$-adic field $K$ with the standard topology inherited from the usual $p$-adic norm $\mid \cdot \mid$. Consider the ultrametric space $X=K^n$ with the topology inherited from the norm $\| ...
0
votes
1answer
173 views

Partitioning the unit ball in an ultrametric space

Consider an ultrametic space $X=K^n$ (for the norm $\| x\| =\max_{i=1,n} (\mid {x_1}\mid,\dots,\mid{x_n}\mid)$ where $K$ is an ultrametric field. Let $B(1):=\lbrace x \in X \mid \|x\| \leq 1\rbrace$ ...
2
votes
1answer
317 views

Kuga-Satake with p-adic methods

Is it possible to construct the Kuga-Satake abelian variety attached to a K3 surfaces (over a local field) only using p-adic methods? If the K3 surface is defined over a local field, the ...
5
votes
1answer
998 views

Definable measure preserving isomorphisms of $p$-adic semialgebraic sets

Hi, Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a ...
5
votes
2answers
1k views

Polynomial reducible modulo every integer

Hi, let $f\in\mathbb{Z}[X]$ be a monic polynomial. Assume that the reduction of $f$ modulo $m $ is reducible for all integers $m\geq 2$. Q1: Is $f$ reducible in $\mathbb{Z}[X]$ ? I've thought ...
7
votes
0answers
386 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 ...
5
votes
1answer
186 views

p-adic noninvariance of dimension

Let $p$ be a prime number. Let $n,m \geq 1$ be such that the topological spaces $\mathbb{Q}_p^n$ and $\mathbb{Q}_p^m$ are homeomorphic. Can we conclude $n=m$? For $\mathbb{Z}_p$ it's false: In fact, ...