# 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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### Has any one seen this sum of roots of unity before?

Fix a prime $p >2$ and $q_1$, $q_2$ such that $q_i - 1$ is exactly divisible by $p$. For any $n$, $a$, $b $, consider the sum
$$\sum_{i=0}^{p^{n-1}-1}\zeta_{p^n}^{aq_1^i+bq_2^i}.$$
Is this always ...

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### question about Sinnott's proof of the Ferrero-Washington Theorem

I'm currently reading the paper "On the $\mu$-invariant of the Γ-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian number ...

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### Analogs of the Weil conjectures for non-archimedian fields

Suppose that $X$ is a smooth and proper variety defined over a perfect non-archimedian valued field $k$ of characteristic $p$. Then one can consider the action of Frobenius on crystalline cohomology. ...

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### The Gamma-transform and $p$-adic $L$-functions

I'm currently reading the paper "On the $\mu$-invariant of the $\Gamma$-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian ...

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### Geometric series in algebraic number fields

For which algebraic numbers $\alpha$ is there a valuation on the number field ${\mathbb {Q}}(\alpha)$ for which the infinite series $\sum_{n=0}^\infty \alpha^n$ converges to $1/(1-\alpha)$?

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### p-adic taylor polynomial [closed]

This might be an easy question but i am sorry for asking this.
Let $f(x)\in\mathbb{Z}_p[x].$ Is it always true that
$$f(x+y)=f(x)+f'(x)y+f''(x)\frac{y^2}{2}+zy^3$$
for some $z\in\mathbb{Z}_p.$ if it ...

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### Hensel's proof that $e$ is transcendental

When he introduced $p$-adic numbers, Kurt Hensel produced an incorrect local/global proof of the fact that $e$ is transcendental. Apparently, the intended proof goes along the following lines: ...

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### Elementary aspects of The Fargues-Fontaine curve

To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius ...

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### Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series

Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...

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### Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...

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### In need of help with parsing non-Archimedean function theory

My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...

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### Decomposition of primes in cyclotomic extensions and their ramifications

Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$.
So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...

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### Difficulty about Jordan decomposition, (and also an ambiguity about the quadratic forms in indecomposable Jordan components of quadratic modules)

I am trying to understand a concept through solving some exercises, but I can't solve one of them, and I need a hint and guide.
I asked my questions in the boxes (See the end of this question). (I ...

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### Transcendentality of Coleman integral

I wonder if there's any work that considers the algebracity/transcendentality of Coleman integral (over $\mathbb{Q}_{p}$). The reason I think about this is because, for hyperelliptic curves, there are ...

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### What is the preimage of the maximal ideal under certain exponential functions?

I'm taking a shot in the dark with this question, so I apologize if it makes no sense.
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $K_n$ be the field obtained by adjoining the $n$-th ...

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### p-adic density of the image of a polynomial

Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. Recently, a user of MO proved that the limit
$$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\...

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### Existence of "nth root function" which is analytic

Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...

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### Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...

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### Bernoulli distributions and $p$-adic measure on $K$

The $p$-adic field $\mathbb{Q}_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can ...

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### A p adic limit of a binomial coefficient

Let $0 \leq a \leq p^n$ be a number coprime to p. Consider the following sequence of binomial coefficients:
$$B_k = \binom{p^{n+k}}{p^ka} $$
as $k\to \infty$. If I did the computation right, the p-...

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### Local global principle for a system of polynomial equations

Suppose $T$ be a system of polynomials homogenous of degree 2 solvable over $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. So, can we claim that $T$ is solvable over $\mathbb{Q}$? I think as of ...

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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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### Computing the ring of power-bounded elements in an affinoid algebra

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $A$ be an affinoid $K$-algebra, i.e. $A$ is isomorphic to a quotient of the Tate algebra $K\left<T_1,\dotsc,T_n\right>$ for some $n$. ...

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### Extension of hyperderivatives

Let $a$ be algebraic over $K:=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. Can one extend continuously the hyperderivatives on $K(a)$?
Recal that the hyperderivative $D_h$ over $K$ is defined by
...

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### $C^*$-algebras over an extension of $\mathbb{Q}_p$?

I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...

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### 'Spherically complete' normed fields

A non-Archimedean normed field $K$ is said to be spherically complete if every decreasing sequence of closed balls in $K$ has non-empty intersection. I am a little puzzled as to why this definition is ...

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### Automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers?

I am interested to see what is currently known about the automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-...

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### A "multi-adic" absolute value / topology?

Let $S$ be a set of finitely many prime numbers. Then, define $\left|\cdot\right|_{S}:\mathbb{Q}\rightarrow\left[0,\infty\right)$ by: $$\left|x\right|_{S}\overset{\textrm{def}}{=}\prod_{p\in S}\left|x\...

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### Image of the ghost map of $p$-typical Witt vectors and $A$-ring structure of $W(A)$

For all ring with unit element $A$ let $W(A)$ be the ring of $p$-typical Witt vectors. Denote by $$\phi\;:\;W(A)\to A^{\mathbb{N}}$$
the ghost map, which is given by
$$\phi(a_0,a_1,a_2,\ldots)\;=\;(\...

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### Power of $2$ dividing a specialized Mittag-Leffler polynomial

While studying the so-called Mittag-Leffler Polynomials, denoted $M_n(x)$, I was looking into the sequence $\frac1{n!}M_n(n)$ which takes the following form
$$a_n:=\sum_{k=1}^n\binom{n-1}{k-1}\binom{n}...

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### How to estimate the highest power of 2 in the partial sum of 2-adic $\log(-1)$ (i.e. $\sum_{i=1}^n\frac{2^i}{i}$)?

The estimate I wanna get is $$v_2(\sum_{i=1}^n\frac{2^i}{i})\geq\min_{t\geq n+1}\{t-v_2(t)\}\tag{*}$$
where $v_2$ is the 2-adic valuation, that is the highest power of 2 defined on $\mathbb{Q}$.
Set $$...

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### Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}_K$. Consider the field $K_\infty$ obtained by adjoining to $K$ all the solutions ...

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### 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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### 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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### 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 ...

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### 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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### 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 ...

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### 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 ...

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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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### 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&...

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### 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 ...

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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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 ...