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.
1,525
questions
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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 ...
3
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0answers
77 views
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 ...
4
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0answers
27 views
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}...
2
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0answers
102 views
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 ...
4
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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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2answers
65 views
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
$\Vert1-0.9999...\Vert=\Vert\lim_{n\rightarrow \infty}\frac{1}{10^n}\Vert=\lim_{n\rightarrow \infty}10^n\rightarrow\infty$.
...
1
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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 ...
5
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1answer
83 views
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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0answers
11 views
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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0answers
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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69 views
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 $...
0
votes
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 ...
2
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1answer
43 views
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
24 views
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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1answer
50 views
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 ...
3
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1answer
71 views
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 $$D= \{(x,w)\in\...
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0answers
27 views
How can i show $\mathbb{Z}_p/p^n\mathbb{Z}_p \approx \mathbb{Z}/p^n\mathbb{Z}$
I would like to show $\mathbb{Z}_p/p^n\mathbb{Z}_p \approx \mathbb{Z}/p^n\mathbb{Z}$.
I have this proof but i dont know how can i finish it:
I know $\forall x\in\mathbb{Z}_p$ $\exists !(\alpha_n)$...
2
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2answers
40 views
Why can we write $\mathbb{Z}_p$ as the disjoint union of balls $a+p\mathbb{Z}_p$
In this chapter, Example 2.1 notes that we can write $\mathbb{Z}_p$ as $$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|...
3
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2answers
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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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1answer
31 views
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 ...
0
votes
1answer
51 views
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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0answers
18 views
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) ...
4
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0answers
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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}$...
4
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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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0answers
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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
40 views
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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0answers
48 views
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 ...
3
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2answers
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 ...
2
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0answers
45 views
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 ...
2
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0answers
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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2answers
50 views
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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2answers
43 views
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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0answers
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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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0answers
73 views
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
54 views
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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6answers
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 $n\leq 1000$...
3
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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'...
3
votes
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 $n\leq 1000$. ...
1
vote
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 ...
1
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0answers
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 ...
3
votes
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 ...
0
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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 ...
