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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3
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0answers
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)}...
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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 ...
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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,...
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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 \...
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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
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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 ...
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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 $\mathbb Q$-...
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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).
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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 ...
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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
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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
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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
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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
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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
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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,...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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-...
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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
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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
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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
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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 ...

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