Questions tagged [p-adic-analysis]
p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.
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$p$-adic numbers in physics
As far as I know, in modern physics we assume that the underlying field of work is the field of real numbers (or complex numbers). Imagine one second that we make a crazy assumption and suggest that ...
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An unfamiliar (to me) form of Hensel's Lemma
In his very nice article
Peter Roquette,
History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355,
Fields Inst. Commun., ...
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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
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=0}^{n-1}\frac1{{n-1\choose k}}.$$
The ...
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Crux of Dwork's proof of rationality of the zeta function?
As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
user97565
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Special values of $p$-adic $L$-functions.
This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.
My understanding is that nowadays there are conjectures which essentially ...
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Automorphisms of $\mathbb C_p$
I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...
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Elementary results with p-adic numbers
I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience ...
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$p$-adic integrals and Cauchy's theorem
A short version of my question is: Is there a $p$-adic theory of integration?
Now let me expand a little further. In introductory texts such as Koblitz' book $p$-adic numbers,.. a bunch of $p$-adic ...
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Ehresmann's theorem over the $p$-adics
I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie ...
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2-adic Coefficients of Modular Hecke Eigenforms
Suppose that $N$ is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level $\Gamma_0(N)$.
For such an eigenform $f$, the coefficients generate (an order in) the ring of ...
user631
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$p$-adic Bott periodicity?
The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...
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When does a p-adic function have a Mahler expansion?
Let $f: \mathbb{Z}_p \rightarrow \mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n \in \mathbb{C}_p$ with
$$
f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}.
$$
...
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Theory of C* algebras over other fields than the complex numbers
How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...
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is there a p-adic implicit function theorem?
I am trying to find a good reference for a version of the implicit function theorem over $p$-adic manifolds. None of the texts I have consulted ( including "$p$-adic numbers, $p$-adic analysis, and ...
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Connectifications?
Like many of my questions, this question is actually aimed at $p$-adic analysis.
One of the main obstacles in doing analysis $p$-adically ist that the $\mathbb{Q}_p$ is totally disconnected.
From ...
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Can perfect numbers be seen $p$-adically?
It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M_{q}:=2^{q}-1$ a Mersenne prime.
As the very defining property of such a perfect number is to fulfill the ...
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Are maps corresponding to affinoid subdomains flat in the Banach sense?
$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$
Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid ...
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Free subquotient of Galois representations coming from Hida theory
Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{...
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On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
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Nijmegen 1978 $p$-adic analysis proceedings
Anyone knows if there is a chance of getting a copy of the following:
Proceedings of the Conference on p-adic Analysis.
Held in Nijmegen, January 16–20, 1978. Report, 7806. Katholieke Universiteit, ...
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A definition of a (amalgamated) direct sum
I am wondering about a definition of a direct sum in page $31$ of this paper by R. Liu.
I am following the notations in page $31$ of the above paper. Let $V$ be a crystalline irreducible ...
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Is there a multivariate analog of Dwork's theorem?
$f(x)=\sum_{i=0}^\infty a_ix^i$ with $a_i\in\Bbb Q$. Let $S$ be finite set of places of $\Bbb Q$ such that:
1. $\forall p\notin S$, $|a_i|_p\leq1$ $\forall i\geq0$.
2. $\forall v\in S$, $f(x)$ ...
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Tweaking the Catalan recurrence and $2$-adic valuations
Among many descriptions of the Catalan numbers $C_n$, let's use the recursive format $C_0=1$ and
$$C_{n+1}=\sum_{i=0}^nC_iC_{n-i}.$$
Then, the $2$-adic valuation of $C_n$ is computed by $\nu_2(C_n)=s(...
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