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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questions
3
votes
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207 views
A question on the Robba ring
Notation is as in the question:
https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring.
We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...
11
votes
1answer
302 views
Open immersion of affinoid adic spaces
If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...
1
vote
0answers
116 views
A p-adic analogue of a result due to Kirillov
Let $k$ be a non-Archimedean local field with char$(k)=0$.
Let $N$ be the group of $k-$rational points of a unipotent algebraic group defined over $k$.
It is known that $N$ is a locally compact and ...
11
votes
1answer
552 views
Where should I learn about the p-adic L-functions of elliptic curves?
Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but ...
2
votes
0answers
38 views
Name for a logarithmic ratio of roots
I'm trying to find a name for the following quantity that came up in my research. I've asked some people and looked around myself but can't find a name, yet it seems like something that has probably ...
2
votes
0answers
63 views
A sequence modified from the Catalan recurrence and $2$-adic valuations
Earlier, I posted this MO question to which Max Alekseyev found counter-examples. I realized that there was some error in my computational programming with Maple. Oh, well $\dots$ I have scaled down ...
3
votes
1answer
169 views
Tweaking the Catalan recurrence and $2$-adic valuations
Among many descriptions of the Catalan numbers $C_n$, let's use the recursive format $C_0=1$ and
$$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(...
20
votes
1answer
1k views
Condensed criterion for sheafiness of adic spaces
Multiple times in talks about condensed mathematics (e.g. the Masterclass talks, Clausen's RAMpAGe talk), it is stated that the derived structure sheaf given by the condensed formalism "fixes&...
2
votes
2answers
165 views
A p-adic logarithm as a limit of discrete logs
I've been searching for something similar to the argument below for about a week now and I just must be missing out on the right key words. Can someone point me in the right direction and/or let me ...
8
votes
1answer
613 views
On the definition of the etale site of an adic space
I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First ...
6
votes
1answer
401 views
What is good $t$-adic like topology on $\mathbb{C}(t)$?
Each function $f\in\mathbb{C}(t)$ can be rewritten in the form $f = a_{k}t^{k}+\ldots+a_0+a_1t+\ldots$, $k\in\mathbb{Z}$ and it is possible to define the topology with the open prebase at zero
$V_{n,v,...
2
votes
0answers
64 views
How to analyze the roots of two variable $p$-adic power series $f(x,y)$ in the maximal ideal $m$?
Let $K \subseteq \mathbb{Q}_p$ be the $p$-adic field, $O_K$ be the ring of integer and $m$ be the maximal ideal.
Consider the single variable $p$-adic power series $f(x)=\sum_{n=0}^{\infty}a_n x^n \...
5
votes
0answers
145 views
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series?
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series ?
We know that if $f(x)=\sum_{i=0}^{\infty} a_ix^i$ be a power series over $p$-adic field, then the Newton ...
3
votes
1answer
67 views
Are injective analytic maps between non-archimedean spaces open?
Let $\Omega$ be a non-archimedean complete field, $n\in\mathbb N$ and $f:\Omega^n\to\Omega^n$ be an injective analytic map.
Is the application $f$ open?
In the complex case, this is a consequence of a ...
1
vote
0answers
70 views
Doubt regarding invariance of discrete absolute value under automorphism
I have been reviewing some basic algebraic number theory and $p$-adic analysis, and the following thought crossed my mind: if $F/ \mathbb Q$ be a finite Galois extension, and $\eta$ be a $\mathbb Q$-...
1
vote
0answers
96 views
I'm looking for a proof of Polya-Bertrandias Theorem
I'm looking for a proof of Polya-Bertrandias rationality criterion in english (not the one from Amice).
0
votes
0answers
64 views
Are analytic functions in several variables open mappings?
It is well known that an analytic function $f$ defined on an open set $\Omega$ is an open mapping: for every open subset $U$ of $\Omega$, $f(U)$ is an open set of $\mathbb C$. Is this result still ...
2
votes
0answers
93 views
What is the p-adic Plancherel measure?
What I know as the Plancherel measure for a group is a measure on the spectrum of $G$, aka the set of irreducible representations - at least for finite groups, this makes perfect sense.
Now, this ...
3
votes
1answer
113 views
Analytic p-adic functions that take an algebraic value
Suppose it exists $r\in\mathbb R$ such that the non constant p-adic function $f(z)=\sum_{n\ge0}a_nz^n$ ($a_n\in\mathbb C_p$) is defined on $\mathcal D=\{z\in\mathbb C_p\mid v_p(z)>r\}$. Does it ...
2
votes
0answers
58 views
Zeroes of the Euler series
Consider a prime $p$. Let $f$ be the Euler series defined by $f(z)=\sum_{n\ge0}n!z^n\}$. It is defined and analytic over $\mathcal D=\{z\in\mathbb C_p\mid v_p(z)>-\frac1{p-1}\}$. I try to check if ...
6
votes
2answers
566 views
Zero of the exponential p-adic
Consider the $p$-adic exponential defined over $\mathbb C_p$. One knows $\exp$ is analytic in the domain $\mathcal D=\{z\in\mathbb C_P\mid v_p(z)>\frac1{p-1}\}$. Does it exist an element $z_0\in\...
3
votes
0answers
104 views
Composition in function fields
Let $k=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. One has the map: $\circ:k\times\{v\in k\mid\deg(v)>0\}\to k$ defined by $f\circ g=\sum_{n\ge-m}a_ng^{-n}$ where $f=\sum_{n\ge-m}a_n\frac1{T^...
4
votes
1answer
253 views
Identity theorem in $p$-adic geometry/analysis
If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $\mathbb{Q}_p$ is disconnected. The modern approach to this is,...
9
votes
0answers
362 views
Full measure properties for Zariski open subsets in $p$-adic situation
Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset ...
3
votes
0answers
171 views
A.C.M. van Rooij's *Non-archimedean functional analysis* (1978) is very out-of-print! Anyone know of any good alternatives?
(This is a literature/reference question.)
So... long story short:
(1) In my present research, I needed a theory of continuous functions from the $p$-adic integers to the $q$-adic integers. Unable ...
4
votes
1answer
214 views
Deformations of the Riemann zeta function
Consider the Dirichlet series (for fixed $0 < a \leq 1$):
$$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$
which reduces to the Riemann zeta function for $a=1$. What is known about this function, ...
2
votes
1answer
248 views
Local to global principle for a pair of bilinear equations?
Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations
$$
A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C
$$
$$
B_{1, 1} x_1 y_1 + B_{...
9
votes
1answer
279 views
Lindemann theorem for Artin-Hasse exponential
Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
2
votes
1answer
153 views
Reduced complete Tate ring which is not uniform?
Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
5
votes
1answer
208 views
Classification of finitely generated modules over non-commutative rings
Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...
4
votes
0answers
84 views
Existence of a “p-adic Mahler measure” or alternatively, the converge of a p-adic sequence
Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence:
$$a_n = \frac{1}{p^{n-1}}\...
1
vote
2answers
196 views
Connection between Volkenborn integral and Haar measure on $\mathbb{Q}_p$
This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic.
Since $\mathbb{Q}_p$ is a locally compact ...
6
votes
0answers
292 views
Numerical analysis with p-adic numbers
How should one go about doing numerical analysis with $p$-adic numbers?
By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian ...
1
vote
0answers
96 views
compact $p$-adic Lie group can be embedded into $O_K^n$ or $\text{GL}_n(K)$?
Let $K$ be a local field of charecteristic $0$ and $G$ be a compact $p$-adic Lie group of dimension $n$, then can $G$ be embedded into $O_K^n$ or $\text{GL}_n(K)$ as a closed subgroup? This is a dual ...
5
votes
0answers
260 views
Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics
Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$.
What can be said about an analytic continuation "in the form of Mittag-...
4
votes
2answers
358 views
The formula for (and computation of) the inverse p-adic mellin transform
So, after scouring the entirety of the internet, I managed to find one (and, so far, only one) source that actually explains how to invert the $p$-adic mellin transform:
$$\mathscr{M}_{p}\left\{ f\...
3
votes
1answer
150 views
Asymptotic analysis using the p-adic Mellin Transform?
In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...
1
vote
0answers
62 views
valuation of a derivative in a completion
Let $q$ be a power of a prime $p$ and $w$ be an irreducible polynomial of $\mathbb F_q[T]$. Denote by $\mathbb C_w$ the completion of an algebraic closure of $K_w$, the completion of $\mathbb F_q(T)$ ...
6
votes
1answer
270 views
Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$
Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to ...
3
votes
1answer
131 views
How does an analytic space correspond to a $p$-adic Banach space
Let $K$ be a finite extension of $\mathbb{Q}_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is ...
3
votes
0answers
133 views
Jacobian change of variables formula for $p$-adic valued integration?
Let $k$ be a $p$-adic field. It's possible to make sense of the Haar measure $\mu_{\operatorname{Haar}}$ on $k^n$ as a $k$-valued measure and define integrals
$$\int\limits_{k^n} f(x_1, ... , x_n) d\...
4
votes
2answers
285 views
$p$-adic series bounded if and only if it has finitely many zeros
Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\...
2
votes
0answers
168 views
Rigid analytic geometry and Tate curve
I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...
0
votes
0answers
51 views
Comparison of growth of entire functions in a $p$-adic field
Let $p$ be a prime number, $\Omega_p$ be the spherically complete extension of $\mathbb C_p$. Consider two entire functions on $\Omega_p$: $f(z)=\sum_{n\ge 0}a_nz^n$ and $g(z)=\sum_{n\ge0}b_nz^n$. ...
6
votes
0answers
96 views
ultrametric Rademacher theorem
The classic Rademacher theorem roughly states that Lipschitz continuous functions are almost everywhere differentiable. However, there are well-known ultrametric counterexamples, see Kobliz's classic ...
2
votes
1answer
118 views
Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$
Let $F$ be the spherically complete extension of $\mathbb C_p$ and $(a_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ such that for all $r\in\mathbb R$, one has $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ ...
7
votes
1answer
229 views
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 ...
2
votes
0answers
198 views
Fontaine - Wintenberger field of norms and imperfect case
Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
6
votes
1answer
395 views
An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$
Let $k$ be a complete non-archimedean field and let $\varphi \colon X \to Y$ be a morphism of rigid analytic spaces over $k$, where $\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$ is affinoid. ...
0
votes
1answer
438 views
p-adic expansion of roots of unity [closed]
Let $w$ be an n-th root of unity, I have two questions
1) What are the conditions on the prime $p$ such that $w\in \mathbb{Z}_p$, and if it is the case what is the p-adic expansion of an n-th root of ...
