# Questions tagged [p-adic]

The p-adic tag has no usage guidance.

85
questions

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### p-adic number field $Q_p $and algebraic numbers [closed]

As we all know, the complex number field $C$ be a finite Galois extension field of the real number field that contains all algebraic numbers.
I want to know the proof of the following proposition:
Any ...

2
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0
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168
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### $G_K$-fixed points of sections of affinoids on the Fargues-Fontaine curve

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$ be its absolute Galois group. There are the Fargues-Fontaine analytic curves $Y_{FF}$ and $X_{FF}$ associated ...

19
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515
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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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### A local model of a Shimura variety and a local Shimura variety

I have a question about the book on p-adic geometry by Scholze and Weinstein.
There are two ‘local theories of Shimura varieties’ written in it.
The one is a local model of a Shimura variety. This is ...

3
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1
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### Algebraic numbers in all $\mathbb Q_p$ [duplicate]

Do there exist non-rational algebraic numbers that belong to $\mathbb Q_p$ for all prime $p$? If yes, can one characterize them?
I spent several days for the first question, and I found nothing. The ...

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304
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### Question about log and exp of a formal group law

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $F$ be a Lubin–Tate formal group law defined over $K$ with endomorphism $f(T)$ corresponding to $\pi$ (a uniformizer of $K$). Then one can define ...

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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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### 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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625
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### Vector bundles on the various sites of a preperfectoid

Let $X$ be a preperfectoid space over $\mathrm{Spa}(\mathbb{Q}_p,\mathbb{Z}_p)$. It has several associated sites, with successively finer topologies: $$X_{an} \subset X_{et} \subset X_{proet} \subset ...

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### Looking for an electronic copy of Huber's Bewertungsspektrum und rigide Geometrie

Lately I've been trying (and have failed) to find an electronic copy of Huber's Bewertungsspektrum und rigide Geometrie, which (from what I understand) is the original reference developing the basics ...

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188
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### Characters of p-adic units

Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...

4
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415
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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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### 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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### Homology of a fiber as a cotorsion product

Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let
$\...

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### Vector bundles on adic spaces

Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? ...

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### Is $K^\times/ F^\times$ compact for local fields?

Let $K/F$ be a finite extension of local fields (of characteristic 0). Is it true that the quotient group $K^\times/ F^\times$ is always compact?
I understand that if the extension is cyclic, it is ...

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### Continous morphisms of a local field with conditions in positive characteristic

Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...

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### On the paper "Patching and the p-adic local Langlands correspondence"

$\DeclareMathOperator\GL{GL}$The question is about the paper: Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin - Patching and the $p$-adic local Langlands correspondence. Let $F$ be a finite ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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^...

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### In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?

This question is inspired from the post linked below:
Can an algebraic number on the unit circle have a conjugate with absolute value different from 1?
What I am curious about is the following: let $\...

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1
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260
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### Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...

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### Freeness of completed homology over universal deformation ring

In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...

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### irrationality of Bessel function in $p$-adic

Let $J_0(z)=\sum_{n\ge 0}\frac{(-1)^n}{n!^2}\left(\frac z2\right)^{2n}$ be the Bessel function considered in $\mathbb C_p$. Let $\alpha\in\mathbb Q^*$ be in the convegence disk of $J_0$. Is $J_0(\...

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114
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### continuous isomorphism of $p$-adic field

In Cassels-Frolich, one can read this theorem (page 57): Let $K$ be a finite finite separable extension of the valued field $(k,v)$. Let $\overline k$ be the completion of $k$ and $(K_j)_{1\le j\le r}$...

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### 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 $g(z)=\sum_{n\ge0}b_nz^n$. ...

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### A complete Tate Huber ring is Banachizable (maybe not)?

I have questions of technical nature.
A complete Tate Huber ring is a complete topological (commutative) ring $A$ admitting an open subring $A_0$ whose topology is the $\varpi A_0$-adic topology, for ...

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### 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 $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ ...

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### P-adic functions on annuli

It is known that a complex analytic function defined on an annulus, say, takes its maximum on the boundary. Does an analogue hold for $p$-adic analytic functions?
More precisely suppose we have a ...

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### Extension of $\mathbb C_p$?

Let $K/\mathbb Q$ be a finite algebraic extension. Consider a prime $p$ and $v$ be a valuation of $K$ above the valuation $v_p$ of $\mathbb Q_p$. Denote by $K_v$ a completion for the valuation $v$ of $...

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### Extension of valuations and completion

Let $L/K$ be a finite extension of fields where $K$ is a number field. Let $v$ be a valuation of $K$ and denotes by $K_v$ its completion for this valuation. One denotes $\tilde v$ be the valuation $...

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### Mixed characteristic in symplectic geometry

Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry?
There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...

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### Automorphisms of $\mathbb C_p$ with constraints

In Automorphisms of $\mathbb C_p$, K. Conrad showed that there exist uncountably many $\mathbb Q_p$-automorphisms of $\mathbb C_p$. I have a quite similar question.
Let $(a_n)_{n\in\mathbb N}$ and $(...

3
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1
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304
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### Analytic continuation of a $p$-adic function

Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences of $\mathbb Q_p$ such that the function $f:z\in\mathbb Q_p\to\sum_{n\ge0}a_nz^n+b_nz^{n+1}$converges in $\{|z|_p<1\}$. Assume ...

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### Convergence of a $p$-adic series

Let $K$ be a local field of characteristic $0$ with valuation $v$. I think
$$\lim_{\substack{q\in K\\q\to1}}\sum_{n\ge0}\prod_{j=1}^n\frac{q^j-1}{q-1}$$ converges to $\sum_{n\ge0}n!\in K$ but I did ...

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### Invariant compact in division ring

Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...

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### Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$
be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\...

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1
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### Centraliser of an absolute Galois group

Let $K$ be a finite extension of $\mathbb{Q}_p$.
Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ?
If yes, how can I ...

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1
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### p-adic L-functions for (dual of) fine Selmer Groups

If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, ...

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### How to deduce Bernstein-Zelevinsky classification from the Langlands one

I am trying to understand the Langlands classification. To that end, I am trying to find how I could deduce the Bernstein-Zelevinsky classifcation from the second description of the Langlands ...

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### Arithmetic of Cuspidal Reps. Fundamental non split stratum and simple stratum

I started to read Colin Bushnell's notes on this title. The last theorem in the 3rd section claims that if $\pi$ is an irr. smooth rep. of $GL_N(F)$ containing a non-split fundamental stratum, then $\...

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### History of Fargues-Fontaine curve

In this paper, Pierre Colmez wrote about some history of the Fargue-Fontaine curve. In this schedule of London Number Theory Study Group, Fargues was said to give a talk on November 15th on " Where ...

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### Automorphisms in function fields with $\mathbb F_q[T]$ globally invariant

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q((1/T))$ for the valuation $-\deg$. Does there exist uncountably many automorphisms of $\Omega$ that let $\mathbb F_q[T]$ stable ...

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### Crystalline cohomology over arbitrary perfect ground fields

In preparing a set of notes, I realize there is a simple question for which I cannot find an easy answer; likely due to simple ignorance. Any introductory text on etale cohomology notes that ...

5
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165
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### $p$-adic valuation of $P_n(p)$ if $P_n$ is the $n$th Legendre polynomial

In 3-adic valuation of a sum involving binomial coefficients the $p$-adic valuation of $P_n(p)$ has been obtained.
Computations indicate that $M_n(p)=\sum\binom{n}{k}^2(p-1)^k$ for odd $p$ has the ...

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547
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### 3-adic valuation of a sum involving binomial coefficients

Let $$a(n) = \sum_{0 \leq k \leq n} {n \choose k}{{n+k} \choose k},$$ and define
$b(n) = \nu_3 \bigl(a(n)\bigr)$, where $\nu_3$ is the $3$-adic valuation. About twenty years ago or so, I discovered (...

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363
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### Cyclotomic Extension of a Perfectoid Space

Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...

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239
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### Specialisation of Z_p[[S_1,...,S_n,X_1,...,X_d]]

Let
$R \colon= {\Bbb Z}_p[[S_1,...,S_n,X_1,...,X_d]]$
be the $(n+d)$-variable formal power series ring over ${\Bbb Z}_p$. Choose an arbitrary ideal ${\frak b}$ of $R$ such that
${\mathrm{ht}}({\...