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Questions tagged [p-adic-number-theory]

In mathematics the $p$-adic number system for any prime number $p$ extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

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What is the meaning of $\rho \nu_{\rho}$?

I am reading the paper. On page 14, section 3.1, I am trying to understand the notation $\rho \nu_{\rho}$, where $\rho$ is a supercuspidal representation $GL_n(F)$, $F$ is a non-archimedean local ...
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In what sense is the L2 norm the canonical norm on $\mathbb{R}^n$ but the max-norm is the canonical norm on $\mathbb{Q}_p^n$?

I understand that the L2 norm is natural to consider on $\mathbb{R}^n$ both because of its geometric intuition, and because it is induced by the dot product on $\mathbb{R}^n$. I also understand that a ...
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13 views

chain of inequalities with $p$-adic anaysis

I'm reading a paper of recurrences sequences. In the Lemma 2 there is a chain of inequalities, which i've locked in the blue box in the image, which I do not understand how they are justified. If ...
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1answer
34 views

Making the subgroup of units of a topological ring a topological group

I'm a beginner in the theory of topological algebra (I'm reading something about it in Robert's "A course in $p$- adic analysis"). At page 24, the author states that if $A$ is a topological ring, the ...
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1answer
30 views

$\Bbb{Z}_px\subseteq\overline{\Bbb{Z}x}$

I'm studying $p$-adic integers and in the proof of the fact that closed subgroups of the additive group $\Bbb{Z}_p$ are ideals (see Robert's "A course in $p$-adic analysis", pp.23) I've found the ...
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2answers
55 views

About summation with p-adic valuation

Here is a problem i came across. Prove that $$\sum_{i=1}^{n}\frac{1}{i}$$ is not an integer for $n \geq 2$. The book olympiad number theory by Justin Stevens says that after writing $\frac{1}{i}$ as ...
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1answer
28 views

Additive characters of $\mathbb{C}_p$

Consider $x \in \mathbb{C}_p$ with $|x|<1$ then for $a \in \mathbb{Z}_p$ we have the characters $$ a \mapsto (1+x)^a $$ where $(1+x)^a= \exp(a\log_p(1+x))$ My question is : it's possible to ...
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1answer
73 views

Lubin-Tate formal groups are $p$-divisible groups

I am trying to understand how to see whether a given formal group is $p$-divisible. Let $A$ be a complete noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ of ...
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1answer
52 views

Looking for this article by Paulo Ribenboim

While going through a forum post (http://mathforum.org/kb/message.jspa?messageID=40112), I found the following paper mentioned: 87a:12014 12J10 13A18 Ribenboim, Paulo (3-QEN) Equivalent forms of ...
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24 views

Equality with $p$-adic analysis

I'm reading a paper of recurrences sequences and I could not understand why the equality mentioned in the underlined lines in the image. I think that it has to do with $p$-adic analysis. If someone ...
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87 views

Norms on fields

I'm doing an introductory module in number theory, and came across the definition of a norm on a field. It seems to agree with the definition of a norm on a vector space over a field (just view the ...
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1answer
41 views

Set $X(\mathbb{Q}_p)$ of $\mathbb{Q}_p$-valued Points not Empty

My question refers to a step in AriyanJavanpeykar's argumentation in following former thread of mine: https://mathoverflow.net/questions/325014/irreducible-smooth-proper-one-dimensional-schemes-...
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1answer
35 views

On the Newton polygon for Laurent series

I'm stuck with an understanding of what should be the Newton polygon for a Laurent series. I'm reading ''An introduction to G-function" by Dwork and he dedicates only three pages to Newton polygons ...
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22 views

About Bernoulli polynomials

My question is about Bernoulli numbers and Bernoulli polynomials in the $p$-adic context. In general in fact Bernoulli numbers are defined as global object so they do not depend on $p$. If $B_k(x)$ is ...
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45 views

$p$-adic analytic function bounded implies coefficients bounded?

Let $K$ be a complete valued subfield of $\mathbb{C}_p$. Let $\mathcal O=\{z \in \mathbb C_p \colon \vert z \vert \leq 1\}$ be the ring of integers in $\mathbb C_p$ and $\mathfrak m=\{z \in \mathbb ...
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1answer
72 views

Squares in $\mathbb{Z}_p$

Consider the integral binary quadratic form $$f(x,y) = 2Axy+Bx^2$$ with $A,B \in \mathbb{Z}$ different from $0$. In Cassel's book "Rational quadratic forms" page 237 he claims that for $p \neq 2$ ...
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1answer
37 views

Algorithm to compute the Teichmuller character

For a given prime number $p$ (for simplicity, let's assume $p\neq 2$), the Teichmuller character is a character of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ that takes values in the ...
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1answer
45 views

How do I show that this $p$-adic function is surjective modulo $p$

Let $p$ be a prime. Let $f: \mathbb{Z}_p \to \mathbb{Z}_p$ be surjective modulo $p^n, \forall n \in \mathbb{N}$, where $\mathbb{Z}_p$ is the set of $p$-adic integers. If it is given that $g: \{0,1,\...
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79 views

About continuous functions on $p$-adic fields

Consider $K/ \mathbb{Q}_p$ a finite extension of the field of $p$-adic numbers. If for every such an extension $ f_K: K \to K$ is continuous can we extend these functions to $\mathbb{C}_p$? My idea ...
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59 views

Whether the p-adic power series $\sum \frac{(n!)^n}{1+(n!)^{2n}} x^n$ converges p-adically

p-adic convergence: The p-adic power series $\sum \frac{1}{n!} x^n$ is divergent. But what about the p-adic power series $\sum \frac{(n!)^n}{1+(n!)^{2n}} x^n$? Does it converge p-adically? Answer: ...
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1answer
69 views

Understanding the $p$-part of the discriminant of a totally real number field with a single prime above $p$

Let $K$ be a totally real Galois number field, and suppose there is only one prime above $p$, with ramification index $\leq p-1$. If $K_p$ is the completion of $K$ at the prime above $p$, the claim ...
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primitive representation of integers over $\mathbb{Z}_p$

In Cassel's book "Rational quadratic forms" page 235 he claims that the form $$x_1^2 + x_2^2 +5(x_3^2 + x_4^2)$$ primitively represents $3 \cdot 2^{2m}$ over $\mathbb{Z}_p$ for every prime $p$ and ...
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1answer
62 views

Embedding $\mathbb C_p$ into $\mathbb C$ and vice versa…?

I'm reading Washington's book on cyclotomic fields, and he mentions that it is sometimes convenient to embed $\mathbb{C}_p$ into $\mathbb C$ and vice versa. In my mind, $\mathbb C_p$ and $\mathbb C$ ...
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1answer
31 views

Reference request: $p$-adic unit for $p≠2$ is a square in $\mathbb{Q}_p \iff$ its first digit is a quadratic residue modulo $p$.

I'm looking for a book to reference that contains the statement in the title. Many thanks for your help.
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1answer
44 views

On the existence of an algebraically closed field containing other fields

This question arose while I was reading a paper I found in the web. It might be very simple, but I don't know the answer. Let $\mathbb{R}$ be the set of real numbers and $\mathbb{Q}_p$ the set of all $...
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1answer
45 views

Why is the residue field of $Q_p$ isomorphic to $F_p$?

$\mathcal{O}=\{x\in Q_p:v(x)\geq0\}$ is a valuation ring $\mathfrak{M}=\{x\in Q_p: v(x)>0\}$ is the maximal ideal of $\mathcal{O}$ Why is $K=\mathcal{O}/\mathfrak{M}$ isomorphic to $F_p$, the ...
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2answers
29 views

Open balls in $p$-adic numbers.

I am new to $p$-adic numbers and was watching an introductory video about it. At 14:51, he says that $r$ only takes values in the form of $p^n$. However, I don't understand why $r$ must be restricted ...
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1answer
26 views

How can I show that this function is a 1-Lipschitz function?

Given $f: \mathbb{Z}_5 \to \mathbb{Z}_5$ such that for some fixed $N \in \mathbb{N}$ $$f(x)= x + 5^N (-x_N +x_N^2 +3)$$ I want to show that $f$ is 1-Lipschitz such that $f$ is induced with the p-adic ...
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Estimating of Big Omega function

Let $\Omega$ be prime big omega function. (Here is description-https://en.wikipedia.org/wiki/Prime_omega_function). Also let $n$ is composite number. Find as good as you can upper bound of number $...
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130 views

Torsion-free abelian groups, tensor product and $p$-adic integers

I'm studying torsion-free abelian groups and I know (see Fuchs, "Infinite Abelian Groups", vol. $2$, pp $154$) that, if $\mathbb{Z}_p$ is the set of $p$- adic integers and $\mathbb{Z}_{(p)}$ denotes ...
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1answer
78 views

Infinite direct sum of p-adic integers is not p-adic

Studying Bousfield localization I stumbled upon this elementary example: we start with $\mathcal{D}$ the derived category of $p$-local abelian groups and we can consider the Bousfield class of $\...
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Extending absolute values on local fields - what is the 'correct' normalization and the relation to the global theory?

I'm having a tough time figuring out the 'correct' normalization for extending absolute values of local fields. I'm also trying to piece together how this interacts with the global theory, so below is ...
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1answer
27 views

Valuation of $x^\lambda$ in a complete, $p$-valued group

Suppose for $p$ a prime that $(G,\omega)$ is a complete $p$-valued group, $x \in G$ and $\lambda \in \mathbb Z_p$ (the $p$-adic integers). Let $x^\lambda$ denote the unique element of $G$ such that $\...
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1answer
75 views

Can we study representation of $p$-adic group by studying $p$-adic Lie algebra?

While I'm studying about representation theory of $\mathrm{GL}(2)$ over local fields, I found that there's no one talking about $p$-adic Lie algebra. However, for Lie groups over $\mathbb{R}$ or $\...
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25 views

On the Newton Polygon for $p-$adic Power series

I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ ...
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2answers
51 views

$p$-adic power series and its maximum in the unit ball

Let $\mathbb{K}$ be an algebraically closed field with a complete absolute value and denote by $R$ its valuation ring. Consider a power series $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
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108 views

Is there a proof of quadratic reciprocity using $p$-adic numbers?

I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. ...
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1answer
30 views

Why is there no equality in Krasner's lemma?

$|\cdot|_p$ denotes the p-adic valuation throughout. Krasner's Lemma: Let $a, b \in \overline{\mathbb{Q}}_p = $ algebraic closure of $ \mathbb{Q}_p$ with $|b-a|_p < |a_i - a|_p$ where the $a_i \...
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78 views

Supercuspidal representation from compact induction

Note: this is a homework problem. Any hints would be great. Consider the group $G=GL_n(\mathbb{Q}_p)$ and an open subgroup $K$ that is compact modulo center. Suppose we have a smooth representation $\...
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$p$-adic-valuation of an expression involving Bernoulli numbers

Let $p = 43, 67$ or $163$ (three primes such that $h(\mathbb{Q}(\sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression \begin{equation} 1+\...
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3answers
89 views

Unitary Central Character by Schur's Lemma

Consider an irreducible smooth representation $\pi$ of the group $G=GL_n(\mathbb{Q}_p)$ with center $Z$. Does there exist a unitary central character for $\pi$? More precisely, is there a (quasi-)...
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47 views

Matrix coefficients of Supercuspidal representations

Let $F$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$, and let $(\pi,V)$ be a supercuspidal representation of $G=GL_n(F)$. In particular, all its matrix coefficients have compact support ...
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1answer
26 views

if $f$ has Newton Polygon consisting of one segment $(0,0)$ to $(n,m)$ with $m,n$ coprime, then $f$ cannot be factored

Let $f(X)\in 1+ X\mathbb{Z}_p[X]$ have Newton Polygon consisting of one segment joining $(0,0)$ to $(n,m)$ with $m,n$ coprime. I have to show that $f(X)$ cannot be factored as a product of two ...
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55 views

Can anyone help me understand this part of a proof involving inverse limits?

I have very little knowledge on inverse limits, in fact I only started reading about it when I came across it on this particular proof for a theorem on the general criterion of the existence of roots ...
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32 views

How can we have he similar decomposition of $ \mathbb{Q}_p$ and $\mathbb{Q}_p(\zeta_p)$?

We have, $\mathbb{Q}_p=$p-adic field, $\mathbb{Z}_p=$ring of p-adic integers, $\mathbb{Z}_p^{\times}=$multiplicative group of units in $\mathbb{Z}_p$. We have the following decompositions: $\...
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35 views

Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$.

Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
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20 views

How does it conclude that sup-norm is not a field norm?

In the book of $\text{Neal Koblitz}$ on p-adic numbers, p-adic analysis and zeta-function, the following exercise is given: Exercise: Let $V=\mathbb{Q}_p(\sqrt p)$, $ \ v_1=1, \ v_2=\sqrt p \ $. ...
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21 views

The tree of $SL_2$ over a local field

I’m studying Chapter II: $\mathbf{SL}_2$ of Serre’s book “Trees”. In paragraph 1, Serre defines the tree of $SL_2$ over a local field $K$. In particular, he considers the set of $\mathcal O$-lattices ...
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1answer
70 views

$p$-adic supremum of cyclotomic polynomial

Let $p$ be a prime number, and $\Phi_{n}(T)$ be the $p^{n}$th cyclotomic polynomial, which we consider as a function on $\mathbb{C}_{p}$. In Pollack's paper 'On the $p$-adic L-function of a modular ...
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1answer
64 views

Absolute convergence of Fourier series in $\mathbb{Z}_p$

For a $p$-adic number $a \in \frac{n}{p^k}+\mathbb{Z}_p\subset \mathbb{Q}_p$ let $\exp(2i \pi a) =\exp(2i \pi \frac{n}{p^k})$ and $\psi_a(x) = \exp(2i \pi ax)$. Then $$Hom(\mathbb{Z}_p,\mathbb{C}^\...