30
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$p$-adic numbers in physics
For an overview of applications of p-adic numbers in physics I would refer to the Wikipedia and Physics.stackexchange links, and to this nLab entry. Regarding the second question "What is the most ...
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29
votes
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Condensed criterion for sheafiness of adic spaces
Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of www.math.uni-bonn.de/people/...
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25
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$p$-adic numbers in physics
Since the question on physics.stackexchange that was referred to in the comments above has been closed, let me essentially repeat my answer here together with some updates.
I think there are mainly ...
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 ...
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19
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Where should I learn about the p-adic L-functions of elliptic curves?
We know how to attach $p$-adic $L$-function to elliptic curves over $\mathbb Q$ only because we know how to attach them to cuspidal modular eigenforms, and we know
by Breuil-Conrad-Diamond-Taylor that ...
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18
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Geometric series in algebraic number fields
This holds precisely for elements which aren't roots of unity. Indeed, by the product formula we have $\prod_v|\alpha|_v=1$, where the product runs over all (finite and infinite) primes. This shows ...
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13
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Cardinality of ${\mathbb{C}_p}$
Not only does $\mathbb C_p$ have the same cardinality as $\mathbb C$, but the larger field $\Omega_p$, the spherical completion of $\overline{\mathbb Q}_p$, also has this cardinality. Further, one can ...
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13
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On the definition of the etale site of an adic space
Great question!
The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat ...
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12
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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 $\...
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11
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$\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$?
A sequence given by $x_1=a$, $x_{n+1}=a^{x_n}$, where $a$ is a positive integer, is eventually constant modulo every positive integer $T$. This is widely known.
A short proof. We induct in $T$, so ...
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10
votes
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convergent series representation for p-adic complex numbers
The blog post http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners gives a good overview of the situation.
To briefly summarize (extracted from the above post — any errors are ...
- 1,856
10
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Topological dimension of $p$-adic manifolds
$p$-adic numbers are locally compact, Hausdorff and totally disconnected (see this nLab page), hence they are zero-dimensional. This means that---at least naively---topological dimension of $p$-adic ...
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9
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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....
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9
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Why $p$-adic measures?
This is extremely late, but hopefully it's still of some use/interest to you.
p-adic L-functions are usually described as 'p-adic interpolations of special values of L-functions'. For Kubota--Leopoldt'...
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9
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$p$-adic analogue of modular forms, upper half-plane, and $L$-functions
The subjects of "p-adic L-functions" and "p-adic modular forms" are so closely intertwined that it's virtually impossible to talk about either one without immediately running into ...
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8
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Nijmegen 1978 $p$-adic analysis proceedings
In the Nijmegen University Repository I only found pages 193-204, Non-archimedean differentiation. Since I presume a copy for private use is OK, I have scanned Amice's and Morita's contributions, you ...
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8
votes
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Can perfect numbers be seen $p$-adically?
(Not a complete answer but a bit too long for a comment.)
There's a fundamental difficulty here in proving the sort of result you envision. If there were some prime $p$ which had to divide every odd ...
- 6,969
8
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Examples of non-splittable norms
All norms on finite-dimensional spaces over $K$ are splittable if, and only if, the field $K$ is maximally complete. You can find this result (and much more) in the paper by Boucksom–Eriksson “Spaces ...
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8
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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|\!| := \...
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7
votes
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A definition of a (amalgamated) direct sum
The key point here is that $\pi(\alpha)$ and $\pi(\beta)$ are isomorphic representations: both of them are the algebraic representation $Sym^{k-2}$ tensored with the smooth representation $Ind_B^G (\...
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7
votes
Direct proof of special case of Hasse's theorem for elliptic curves
Hasse bound for elliptic curves of the form $y^2=x^3+b$ can be proven without algebraic geometry. It was done by Schrutka in the article Ein Beweis für die Zerlegbarkeit der Primzahlen von der Form $...
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7
votes
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Reference Request: Specialization map in Huber's Context
In case you're still interested: Bhatt recently proved that for any Tate-Huber pair $(A,A^+)$, the topological space $\mathrm{Spa}(A,A^+)$ is homeomorphic to an inverse limit of admissible blowups in ...
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7
votes
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Are maps corresponding to affinoid subdomains flat in the Banach sense?
This is not true. Assume that $X$ is the closed unit disc (given by $|T| \le 1$, with algebra $B$) and $V$ is a smaller disc (given by $|T| \le r$ for some $r \in (0,1)$, with algebra $B_V$). Consider ...
- 4,132
6
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'Important' applications of p-adic numbers outside of algebra (and number theory).
The 2-adic numbers have been used in cryptography, particularly in the analysis of feedback shift registers. See e.g.
2-adic shift registers, Klapper and Goresky, Fast Software Encryption 1993
...
6
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What is the value of $p$-adic $\zeta$-function at positive integer point?
Let me mention some irrationality results about p-adic zeta values:
In 2005, Frank Calegari arxiv link proved that for $p=2,3$, the p-adic zeta values $\zeta_p(3)$ is irrational (hence nonzero). It ...
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6
votes
P-adic C* algebras
Me, Alcides Buss, and Devarshi Mukherjee proposed an answer to this question in this paper.
$\textbf{The idea:}$ Note that via GNS, we can view a complex/real $C^*$-algebra essentially as a closed $*$-...
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6
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p-adic representations of $GL_2(\mathbb{Q}_p)$
Apparently you didn't read the references you quoted terribly carefully, since exactly this question is addressed by the footnote on the bottom of page 4 of the paper you link:
"L'application qui ...
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6
votes
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Binomial coefficients in discrete valuation rings
If $p=2=n$, $d=1$, $V=\mathbf{Z}_2[\sqrt{2}]$ and $\pi=\sqrt{2}$, then
$$ \binom{\pi^d}{n} = \frac{\sqrt{2}(\sqrt{2} - 1)}{2} = 2^{-1/2}\cdot \mathrm{unit} \notin V. $$
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6
votes
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How does an analytic space correspond to a $p$-adic Banach space
Without reading much beyond the pages adjacent to page 6, but looking at [ST3] (especially seeing the references [BGR] and [NFA]), I believe Berger-Colmez are using $K_n$-analytic spaces in the sense ...
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6
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Local to global principle for a pair of bilinear equations?
In fact such system of equations always have a solution (at least if the coefficients are general).
Let $X$ denote the closure of your variety in $\mathbb{P}^4_{\mathbb{Q}}$. Explicitly:
\begin{align*...
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