25
votes
Accepted
A funny metric over $\mathbb{N}$
This metric is mentioned in the Encyclopedia of distances (Chapter 10.3), written by Michel Marie Deza and Elena Deza. Here is the relevant paragraph:$\newcommand{\lcm}{\operatorname{lcm}}$
Let $\...
- 8,401
24
votes
Hensel's proof that $e$ is transcendental
You can read Hensel's original article here. The argument is really simple: from this expansion, we see that $e$ satisfies an equation of the form $y^p=1+pu$ where $u$ is a $p$-adic unit. But the ...
- 28.2k
15
votes
Accepted
Centraliser of an absolute Galois group
For an extension $L/\mathbf{Q}_p$, let $G_L$ denote the absolute Galois group $\mathrm{Gal}(\overline{L}/L)$.
If $\sigma \in G_{\mathbf{Q}_p}$ acts centrally on $G_K$, then it also acts centrally on ...
14
votes
Accepted
Is the p-adic Lindemann-Weierstrass Conjecture still open?
Here is a 2018 paper, A Note on One-dimensional Varieties Over the Complex p-adic Field, that still lists the "full" statement as a conjecture; "half" of the statement, meaning that at least $\lfloor ...
- 188k
13
votes
Accepted
Question about log and exp of a formal group law
The radius of convergence of any formal group $F$ (one-dimensional, finite height) is $1$, in other words $L_F$ will converge at all $z\in\Bbb C_p$ with $v(z)>0$.
In particular, the logarithm is ...
- 4,193
12
votes
Accepted
Zero of the exponential p-adic
The exponential function satisfies $\exp \left({x + y}\right) = \exp\left(x \right) \exp\left(y \right)$ for $x, y$ in the convergence domain. It also satisfies $\exp \left( 0 \right) = 1$. So if $\...
- 730
12
votes
Accepted
In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?
Zero is the only algebraic integer which has all its conjugates strictly inside the complex unit circle. (Look at the norm.)
For explicit examples with conjugates on either side of the unit circle, ...
- 2,391
11
votes
Accepted
3-adic valuation of a sum involving binomial coefficients
The $3$-adic evaluation you seek is compactly given by
$$\nu_3(a_{2n})=\nu_3\left(\binom{2n}n\right) \qquad \text{and} \qquad
\nu_3(a_{2n+1})=\nu_3\left(3(2n+1)\binom{2n}n\right),$$
which can be ...
- 43.2k
10
votes
Accepted
P-adic functions on annuli
$\def\bQ{\mathbb{Q}}\def\bF{\mathbb{F}}\def\bZ{\mathbb{Z}}$This is false as stated because of the following important difference between $K$ and $\mathbb{C}$: the former is not algebraically closed. ...
- 7,367
10
votes
Accepted
Convergence of a $p$-adic series
Is this question being asked just out of curiosity? As far as I know, the series $\sum_{n \geq 0} n!$ is not important in $p$-adic analysis.
For $j \in \mathbf N$ and $q \not= 1$, let $(j)_q = (q^j-1)...
- 50.6k
9
votes
Accepted
Algebraic numbers in all $\mathbb Q_p$
(cw answer based on Wojowu's link)
The only algebraic extension of $\mathbf{Q}$ that embeds into $\mathbf{Q}_p$ for all $p$ (or even for a density 1 set of primes) is $\mathbf{Q}$ itself.
For if $P\...
9
votes
Zero of the exponential p-adic
It’s hard to see in what sense it could be true that the exponential function is “defined over $\Bbb C_p$”, since the logarithm is defined on the whole open unit disk there, and has so very many zeros....
- 4,193
9
votes
Convergence of a product in $\mathbb Q_2[[X]]$
The coefficients you use are all in $\mathbf Z_2$, so I advise working in $\mathbf Z_2[[x]]$ rather than in $\mathbf Q_2[[x]]$.
You are using the wrong topology on the power series, as mentioned ...
- 50.6k
8
votes
Accepted
Vector bundles on the various sites of a preperfectoid
There is some relevant work of Ben Heuer on this.
In short, analytic and etale vector bundles agree on all sousperfectoid adic spaces (and much more generally, for all "etale sheafy" adic ...
- 21.3k
8
votes
Vector bundles on adic spaces
The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ ...
- 21.3k
8
votes
Accepted
Maximum modulus principle over the $p$-adic integers
No. Let $K$ be a nonarchimedean discrete valued field with a finite residue of order $q$. Let $f(x) = x^q - x$ and $\pi$ be a uniformizer in $K$ (generator of $\mathfrak m_K$). Then $|\!|f|\!| := \...
- 50.6k
7
votes
3-adic valuation of a sum involving binomial coefficients
Indeed, the observation by Max Alexseyev seems to provide the idea I need!
The conjecture I made above can be generalized as follows:
Let p > 2 be a prime, and let $L_n (x)$ be the Legendre ...
- 3,028
7
votes
A funny metric over $\mathbb{N}$
As indicated by @MartinSleziak the function
$$f(x,y):=\frac{xy}{\gcd(x,y)^2} = \frac{\operatorname{lcm}(x,y)}{\gcd(x,y)}$$
has interesting properties for example as indicated in this question about ...
- 3,044
6
votes
Accepted
Vector bundles on adic spaces
$\newcommand{\cO}{\mathcal{O}}\newcommand{\bZ}{\mathbb{Z}}$Let's first work out the case $\mathcal{E}=\mathcal{O}_X$. We want a space $E\to X$ such that $Hom_X(S, E)=\cO_S(S)=Hom(S,\mathbb{A}^1)$. ...
- 7,367
6
votes
Accepted
Is $K^\times/ F^\times$ compact for local fields?
$O_K^\times$ is compact thus so is $$K^\times/ \pi_F^\Bbb{Z}=\pi_K^{\Bbb{Z/eZ}} \times O_K^\times, \qquad e=\frac{v(\pi_F)}{v(\pi_K)}$$ Being a quotient of a compact group by a closed subgroup $$K^\...
- 3,403
6
votes
Partition of unity for analytic manifolds over non-Archimedean local fields
It follows from Lemma 1 (part (2)) on page 7 in http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bernst_Lecture_p-adic_repr.pdf
In the non-compact case, I think that you need to ...
- 2,639
4
votes
In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?
To expand on GNiklasch's answer, and analyse what you write as well: we always have (when complex conjugation is central in the Galois group) have
$\overline{\alpha^{\sigma}} = {\bar \alpha}^{\sigma}$ ...
- 44.4k
4
votes
Accepted
Looking for an electronic copy of Huber's Bewertungsspektrum und rigide Geometrie
An electronic copy of "Bewertungsspektrum und rigide Geometrie" is available from Wuppertal University.
[year-old post, bumped to the front page by a spammer, answered for the record]
- 188k
4
votes
Accepted
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
At @MikhailBorovoi's request, I copy here two comments 1 2 from p-adic field extension of degree 2n without a subfield of degree 2? as an answer to Question 4. (The comment suggested that they answer ...
- 12.9k
3
votes
Accepted
Image of the ghost map of $p$-typical Witt vectors and $A$-ring structure of $W(A)$
I don't have a full answer, however I also don't have enough reputation to just comment, so I will post this as an answer.
In what follows, we will overline all projections modulo $p$.
Let $A$ be any ...
3
votes
Accepted
Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?
Yes it is. The extension $K_\infty/K$ is deeply ramified. Now apply proposition 6.6.6 of Gabber-Ramero's "Almost ring theory".
- 6,924
3
votes
Accepted
Unramified extension over $ \mathbb{Q}_{p} $
You did not specify $f$: it should be equal to $n$. A particular generator of the resulting cyclic Galois group is the Frobenius automorphism, which acts on the $(p^n−1)$-th roots of unity by $x\...
- 105k
3
votes
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let's say that a [pro]finite group $G$ is $p$-supersolvable if for all [closed] normal subgroups $N_1\subset N_2$ of $G$ with $N_1$ maximal $G$-normal proper subgroup of $N_2$, the group $N_2/N_1$ is ...
- 63.9k
3
votes
A funny metric over $\mathbb{N}$
Not an answer, but I was curious about the "shape" of numbers determined by this metric. Here's a t-SNE plot of the numbers from 1 to 256. As with any projection of this type, the geometry ...
- 6,551
2
votes
Analytic p-adic functions that take an algebraic value
For the first, yes. Without loss of generality by shifting we may assume $a_1 \neq 0$.
For $\alpha\in \mathbb Q$, let $x_0=0$ and $x_{n+1} = x_n + \frac{\alpha-f(x)}{a_1}$. To check that $x_n$ ...
- 148k
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