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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1answer
34 views

Finding an example for an elliptic curve over the p-adics with bad reduction but potential good reduction

Problem I would like to find an elliptic curve $E$ over $\mathbb{Q}_p$ given by the equation $$ E: \quad y^2 = x^3 - 27 c_4 x - 54 c_6. $$ with the following properties: $E$ does not have good ...
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A question on Fontaine's periods rings

Let $K$ be a complete discrete valued field of characteristic zero with perfect residue field $k$ of characteristic $p>0$, $\mathcal{O}_K$ its ring of integers, $C$ the completion of an algebraic ...
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On a tamely ramified extension of $\mathbb{Q}_{p}$

I'm stuck with the following problem given in a book which I'm reading, it's about creating a tamely ramified extension of $\mathbb{Q}_{p}$. Let $p\in\mathbb{Z}$ be a prime number, and let $\mathbb{Q}...
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Canonical lift of Elliptic curve in Smart attack

From the answer to why the Smart attack fails for a particular lift, the answer provided along with the journal version of Smart's paper regarding the smart attack seems to suggest that when the ...
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1answer
67 views

Kernel of $\mathbb{Z}_p \to \mathbb{Z}/p^{n}\mathbb{Z}$ equals to $p^n \mathbb{Z}_p$.

Let $\mathbb{Z}_p$ denotes the ring of $p$-adic integers, i.e., $\mathbb{Z}_p:= \varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. Then consider the projection map $\pi_{n}: \mathbb{Z}_p \to \mathbb{Z}/p^{n}\...
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In p-adic metric, what is the distance between 0.9999… and 1? [closed]

In 10-adic (though 10 is not a prime number) metric, we know that $\Vert1-0.9999...\Vert=\Vert\lim_{n\rightarrow \infty}\frac{1}{10^n}\Vert=\lim_{n\rightarrow \infty}10^n\rightarrow\infty$. ...
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1answer
33 views

Computing ramification in extension of complete DVRs

Assume I am given a finite primitive extension of complete discretely valued fields $L=K(\alpha)/K$, say with monic integral minimal polynomial $f$ for $\alpha$. How does one systematically compute ...
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A theorem of Lutz about the structure of the points of an elliptic curve over a finite extension of $\mathbb{Q}_p$

Reading the article of Greenberg "Iwasawa Theory for Elliptic Curves", he cites (p.13) a theorem of Lutz that says: Theorem: Let $E/K$ be an elliptic curve defined over a finite extension $K$ of $\...
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How do we get the following reduction to prove the Poincaré series is rational?

In the text by Jan Denef on The rationality of the Poincaré series associated to the $p$-adic points of a variety he creates a relation between the Poincaré series $P(T)$ and an integral $I(s)$. The ...
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21 views

What is the $p$-adic Haar measure of this set?

Let $f_i(x1,\ldots,x_m)$ be polynomials in $m$ variables over $\mathbb{Z}_p$, for $i=1,\ldots, r$. Let $N_n$ be the number of elements in $\{x\mod p^n\mid x\in\mathbb{Z}_p^m, f_i(x)=0\text{ for }i=1,\...
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What's the form of the subgroups of $(\mathbb{Q}_p/\mathbb{Z}_p)^2$?

Let $p$ be prime integer. I've read that every subgroup of $(\mathbb{Q}_p/\mathbb{Z}_p)^2$ has the form \begin{equation} (\mathbb{Q}_p/\mathbb{Z}_p)^e\times U \end{equation} with $0\le e \le 2$ and $...
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1answer
23 views

Proof of the $p$-adic Haar measure of $\{w\in\mathbb{Z}_p\mid \lvert w\rvert\leq 1/p^n\}$

The Haar measure on the $p$-adic integers is a measure which is translation invariant, finite for compact sets and positive for sets with non-empty interior. As such we can define it so that the Haar ...
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A reference for the proof of $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$

I read an article where it is said: $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$ where $E$ is an elliptic curve over $\mathbb{Q}_p$ and $E_1(\mathbb{Q}_p)=\{P\in E(\mathbb{Q}_p):\tilde{P}=\tilde{O}\}$. ...
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1answer
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Isomorphism between $H^1(\mathbb{Z}_p, M)$ and $M_{\mathbb{Z}_p}$

Let $\Gamma$ be a multiplicative group isomorphic to the $p$-adic integers $(\mathbb{Z}_p,+)$, and let $M$ be a discrete torsion $\Gamma$-module. Let $\gamma$ be a topological generator for $\Gamma$. ...
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Is $\mathbb{Q}_2$ complete?

Is the $p$-adic field $\mathbb{Q}_p$ complete respect to the $p$-adic norm $|\cdot|_p$ when $p=2$?
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reference for this theorem about elliptic curves over $\mathbb{Q}_p$.

Do you know a reference where i can read the proof of this theorem?: Let $p$ be an odd prime and let $E$ be an elliptic curve over $\mathbb{Q}_p$ for which its reduction $\tilde{E}$ is an elliptic ...
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1answer
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Write this $p$-adic set as boolean combination of simpler sets.

I am studying a paper written by Jan Denef on The rationality of the Poincaré series associated to the $p$-adic points on a variety. In this paper he defines an integral over a set $$D= \{(x,w)\in\...
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How can i show $\mathbb{Z}_p/p^n\mathbb{Z}_p \approx \mathbb{Z}/p^n\mathbb{Z}$

I would like to show $\mathbb{Z}_p/p^n\mathbb{Z}_p \approx \mathbb{Z}/p^n\mathbb{Z}$. I have this proof but i dont know how can i finish it: I know $\forall x\in\mathbb{Z}_p$ $\exists !(\alpha_n)$...
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Why can we write $\mathbb{Z}_p$ as the disjoint union of balls $a+p\mathbb{Z}_p$

In this chapter, Example 2.1 notes that we can write $\mathbb{Z}_p$ as $$p\mathbb{Z}_p\cup(1+p\mathbb{Z}_p)\cup\cdots\cup (p-1+p\mathbb{Z}_p).$$ I understand that $\mathbb{Z}_p=\{x\in\mathbb{Q}_p:|x|...
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Parametrization of the solutions of $a^2+b^2=1$ in the $p$-adic integers

I was trying to solve the equality $a^2+b^2=1$ in $\mathbb{Z}_p$, the $p$-adic integers. If $p =1$ mod $4$ then $\mathbb{Z}_p$ admits an element $i$ such that $i^2=-1$, using this we can define two ...
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Reference Request - Point counting over elliptic curves over $\mathbb{Q}_p$ and characteristic polynomial of their induced Galois representation

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and $\rho = \rho_E : G_{\mathbb{Q}_p} \to \operatorname{GL}_2(\mathbb{C})$ its induced Galois representation. We can define the characteristic ...
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Representation of p-adic integers

Let $U_p:=\{(a_1,a_2,\dots) \in \Bbb Z_p:a_1 \neq 0\}$ and $x \in \Bbb Z_p \setminus \{0\}$. I don't understand why we can represent $x$ in an unique way as $x=p^{v_p(x)}u$, with $u \in U_p$ and $v_p(...
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1answer
26 views

Questtion about Quotient field of a discrete valuation ring.

Let be $K$ a field and $v:K^{*}\longrightarrow \mathbb{Z}$ a discrete valuation. We can suppose $v$ is sobrejective. The set $R=\{x\in K^{*}:v(x)\geq 0\}\cup \{0\}$ is called the valuation ring of $v$...
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When a power series $f(x) \in x O_K[[x]]$ with $f'(0) \not\equiv 0 \mod m_K$?

Let $K$ be the finite extension of $p$-adic field $\mathbb{Q}_p$ with algebraic closure $\bar K$. Let $O_K$ be the ring of integer and let $m_K$ be the maximal ideal. Consider a power series $f(x) ...
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Frobenius between $\mathbb{Z}_p[p^{1/p^m}]$-modules etale

The questions are motivated by P. Scholze's answer in MO question https://mathoverflow.net/questions/132438/why-is-faltings-almost-purity-theorem-a-purity-theorem. Consider for every $m \in \mathbb{N}$...
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1answer
118 views

What is wrong with this application of Ax-Kochen?

I have been bashing my head for a couple of hours trying to find out what's wrong with the following argument: Let $K=\mathbb{Q}_p(p^{1/p-1})$ and $L=\mathbb{Q}_p((-p)^{1/p-1})$. These are two ...
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How do I extract a square root in a field of p-adic numbers? [duplicate]

Retrieval of square root in the field of p-adic numbers. Find ${\sqrt 5}$ in field $\Bbb {Q_{11}}$ Find ${\sqrt 3}$ in field $\Bbb {Q_{13}}$ Any information that will help solve this task will be ...
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1answer
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About the quotient space $Z/p^iZ$.

Fix a prime number $p$ and consider $Z_p=\{(a_i)_{i\in N}:a_i\in Z/p^iZ_p\ and\ \phi_{i+1}(a_{i+1})=a_i\ for\ every\ i \}\subset \prod_{i=1}^{\infty}(Z/p^iZ)$, where $\phi_i:Z/p^iZ\rightarrow Z/p^{i-...
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Is there a p-adic triangle geometry?

While understanding an analytic construction of $p$-adic numbers is relatively easy (as are understanding Hensel decomposition and basic operations — addition, multiplication…), the field seems to get ...
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91 views

Proof by hand that $\mathbf{Q}_p$ is complete

We take the completion of $\mathbf{Q}$ with respect to the p-adic norm in the following way. We take the ring of all Cauchy sequences $C$ modulo the maximal ideal $M$ of all null sequences and we ...
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If $\rho$ is a mod $p$ irreducible representation of $G_{\mathbb{Q}_p}$, why is it tamely ramified?

Fix a prime $p$ and let $\rho:G_{\mathbb{Q}_p}\rightarrow GL_2(\overline {\mathbb{F}}_p)$ be an arbitrary continuous representation. I found the following statement in a paper on non-ordinary modular ...
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55 views

Functions from the integers to themselves that converge in every p-adic topology

Let $f: \mathbb Z\to \mathbb Z$ be an arbitrary function. We can extend it to a function $f_p: \mathbb Z_p \to \mathbb Z_p $ if and only if $f(n) \to f(m) \in \mathbb Z_p$ as $n \to m \in \mathbb Z_p$....
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Standard topology doesn't match with the topology induced by the p-adic metric

I want to show that the standard topology on $\Bbb Q$ doesn't match with the topology that is induced by the p-adic metric. For this I wanna show that for the open set $M:= \{x \in \Bbb Q: |x|_p <\...
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Triangle inequality for the $p$-adic metric

I try to understand the triangle inequality prove for the $p$-adic metric. The proof is given as: $$\DeclareMathOperator{ord}{ord}|x-y|_p = p^{-\ord_p(x+y}\leq p^{-\min \{\ord_p(x),ord_p(y)\}} = \...
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Closure of $Z_{(p)}$ in $Q_p$

I'm reading Number Theory 1 written by Kato, Kurokawa and Saito. I have a question about one of the proposition in chapter2: $p$-adic Numbers. Definitoin : $\mathbb{Z}_{(p)}$: { $\frac{a}{b}$ | $a,b ...
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1answer
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Proof explaination - $\sum_{i=1}^{n} \frac{1}{i}$ is not an integer for $n>1$

I was reading a proof to the following fact: for $n>1$, $\sum_{i=1}^{n} \frac{1}{i} \notin \mathbb{Z}$. The proof is as follows: Denote for prime $p$ by $v_p(a)$ the p-adic valuation of $a$. Write ...
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Proof that (x OR c) = x + c - (x AND c)

Proof that (x OR c) = x + c - (x AND c), where $x$ -- p-adic number in $Z_2$ and $c$ -- is positive integer. AND, OR - bitwise operations
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Logarithm and Lubin-Tate formal group

Let $K$ be a finite extension, by Milne's online note "class field theory", $m_{\mathbb{C}_p}$ has a natural $O_K$ module structure where the action is given by $[a]_f$. For such a $f$, there exists a ...
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What is the measure of the function?

Is the $T$-function $f(x) = (x \text{AND} c) + (x^2 \text{OR} c)$, where $c$ is positive integer ergodic in the space $Z_2$ (p-adic numbers)? What is the measure of this function? I am trying to use ...
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1answer
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Showing that Q is not complete with respect to the 2-adic and 3-adic absolute value

I have seen here how to show that Q is not complete with respect to the $p$-adic absolute value, where $p\geq5$. Is there a similar proof/idea for $p=2$ and $p=3$?
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1answer
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Need help in an argument related to P-adic valuation

I am unable to understand why an argument related only to p-adic number theory must be true . Question: Assume (2.5) equivalent to equation S= -P to simplify notation( Here S and P are sums ...
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490 views

$3^n$ does not divide $4^n+5$ for $n\geq 2$

Question as in the title : does anyone know how to prove that $3^n$ does not divide $4^n+5$ for $n\geq 2$ or find a counterexample ? My thoughts : (1) I have checked that this is true for $n\leq 1000$...
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2answers
106 views

Prove that number of times $3$ divides $2^n\pm1$ is exactly one more than the number of times $3$ divides $n$

TL;DR How to prove the eight congruences at the end of this post? Remark. My number theory is rusty and I'm trying to prove the following observations. Motivation: This result easily implies that $3^...
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1answer
36 views

How to prove that $\mathbb{Z}_p^\times$ is isomorphic to $(\mathbb{Z}_p, +) \times \mathbb{F}_p^\times$?

Deduce that $\mathbb{Z}_p^\times$ is isomorphic to $(\mathbb{Z}_p, +) \times \mathbb{F}_p^\times$. I have already deduced that $(1+p\mathbb{Z}_p, \cdot)$ and $(\mathbb{Z}_p,+)$ are isomorph. But I can'...
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2answers
124 views

$3^n$ does not divide $8^n+1$ for $n\geq 4$

Question as in the title : does anyone know how to prove that $3^n$ does not divide $8^n+1$ for $n\geq 4$ or find a counterexample ? My thoughts : I have checked that this is true for $n\leq 1000$. ...
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1answer
31 views

Can you model p-adic numbers computationally?

Please forgive if the question is not perfectly formulated. The general notion is, is there some way to model - build a representation of - the p-adic numbers, in computer code? For example, could we ...
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0answers
25 views

Vanishing second cohomology with coefficients in a $p$-adic vector space

Let $\Gamma$ be a discrete group, and $V$ a (not necessarily finite-dimensional) $\mathbb{Q}_p[\Gamma]$-module. I am looking for general criteria that ensure that $H^2(\Gamma, V) = 0$. The simplest ...
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41 views

valuation of a derivative

In a lecture, I saw this exercice: Denote by $k$ the field $\mathbb F_q(T)$ of characteristic $p$. Let $H$ be an irreductible polynomial of $\mathbb F_q[T]$. Denote by $v_H$ the $H$ be the valuation ...
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2answers
135 views

Subfield of degree $p$ of $\mathbf{Q}_p(\zeta_{p^2})$

In this answer, Keith Conrad claims that a generator of the subfield of degree $p$ over $\mathbf{Q}_p$ of $\mathbf{Q}_p(\zeta_{p^2})$ is given by $$\sum_{a^{p-1}\equiv 1\bmod{p^2}} \zeta_{p^2}^a.$$ I ...
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1answer
28 views

Finding the number of elements of a quotient group.

For a finite extension $K$ of $\mathbb{Q}_p$ with residue field of order $q$, ring of integers $\mathcal{O}_K$, and prime element $\pi$, define $(1+\pi^n\mathcal{O}_K)$ to be the multiplicative ...

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