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Questions tagged [p-adic-numbers]

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

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Can the p-adic be countable?

Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
Ember Edison's user avatar
4 votes
1 answer
121 views

Maximum modulus principle over the $p$-adic integers

Consider $\mathbb{Z}_p$ the $p$-adic integers. Let $f\in\mathbb{Z}_p[x]$ be an arbitrary polynomial in one variable. Write $f(x) = \sum_{k}a_kx^k$. Is it true that $\|f\|:= \max_k |a_k|_p = \sup_{t \...
Luiz Felipe Garcia's user avatar
1 vote
0 answers
32 views

Time complexity of Magma's `NormEquation` for quadratic extensions of $2$-adic fields

Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...
Sebastian Monnet's user avatar
4 votes
1 answer
500 views

Subgroup of p-adic units

Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product ...
Nandor's user avatar
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2 votes
0 answers
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Is the extension field by zeros of $x^{2m}-p^{2(m-1)}=0$ over $\mathbb Q_p$ totally ramified?

Let $m \geq 2$ be an integer. Consider the polynomial $f(x)=x^{2m}-p^{2(m-1)} \in \mathbb{Q}_p[x]$. I want to study the field extension by the zeros of $f(x)$ over $\mathbb{Q}_p$. What is the degree ...
MAS's user avatar
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4 votes
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields

Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
Sebastian Monnet's user avatar
4 votes
0 answers
173 views

Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
Ashvin Swaminathan's user avatar
3 votes
1 answer
163 views

Approximating $p$-adic power series by polynomials

Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
Ashvin Swaminathan's user avatar
-2 votes
2 answers
127 views

How do you prove that the series 5, 25, 625, ... can be continued forever to give a 10-adic solution to $n^2=n$? [closed]

How do you prove that the series 5, 25, 625, ... can be continued forever to give a 10-adic solution to $n^2=n$? Here's a proof for a different solution (...1787109376): https://oeis.org/A018248/...
Tony Phillips's user avatar
2 votes
0 answers
121 views

p-adic Banach space and complete tensor product

Let $p$ be a prime and $\mathbb{C}_{p}$ the completion of the algebraic closure of the $p$-adic number field $\mathbb{Q}_p$. Let $M$ be a $\mathbb{Q}_p$-Banach space. We denote by $M\mathbin{\widehat{\...
user521844's user avatar
4 votes
1 answer
297 views

Conductor and local Kronecker–Weber theorem

Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...
Yijun Yuan's user avatar
1 vote
1 answer
75 views

Compact subgroups of a linear group over non-Archimedean local field

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
asv's user avatar
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2 votes
1 answer
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How to get a ball in the nonvanishing locus of a polynomial in $\mathbb Z_p[x_1,\cdots,x_n]$ canonically?

Suppose $f\in \mathbb Z_p[x_1,\cdots,x_n]$, and consider $D(f):=\{(𝑥_1,…,𝑥_𝑛)∈ℤ^𝑛_𝑝:𝑓(𝑥_1,…,𝑥_𝑛)≠0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
Richard's user avatar
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3 votes
0 answers
162 views

Algebraic properties of Witt vectors $W(K^{\flat\circ})$, $K$ a characteristic 0 perfectoid field

Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \...
Vik78's user avatar
  • 527
1 vote
1 answer
138 views

Quadratic unramified extension of a p-adic field

Let $F$ be a non-archimedean local field of residual characteristic $p\neq 2$, and let $E=F[\sqrt{\epsilon}]$ be the quadractic unramified extension, here $\epsilon$ is a non-square element of $\...
Ekta's user avatar
  • 63
5 votes
1 answer
226 views

p-adic L functions from Selmer groups - how canonical are they?

For this question, I am going to be very concrete but very much appreciate broader viewpoints. Let $F$ be a number field and define $F_n = F(\mu_{p^n})$ and let us suppose for simplicity that $\mu_p \...
Asvin's user avatar
  • 7,686
1 vote
1 answer
231 views

A vanishing sum and related $p$-adic congruences

Recently I had a curious discovery. Namely, I have made the following conjectures. Conjecture 1. We have the identity $$\sum_{k=0}^\infty\frac{(10k-1)\binom{3k}k\binom{6k}{3k}}{(2k+1)512^k}=0.\label{1}...
Zhi-Wei Sun's user avatar
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1 vote
1 answer
193 views

Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring

Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
HASouza's user avatar
  • 293
5 votes
1 answer
473 views

General algebraic result obtained from consideration on $\mathbb{Q}_p$

There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields. For instance, the fact that a polynomial $P$ admits a ...
Weier's user avatar
  • 151
0 votes
0 answers
68 views

Space of non-archimedean characters is nonempty

Let $k$ be an algebraically closed complete non-archimedean field. Let $\mathcal{O}_k$ be its ring of integers. Suppose that $A$ is a $k$-Banach algebra, and $B$ is its closed unitary ball. Note that $...
Luiz Felipe Garcia's user avatar
-1 votes
1 answer
292 views

Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]

Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$ i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$ Now let $X$ be the restriction of its ...
it's a hire car baby's user avatar
0 votes
0 answers
30 views

Min-norm solution of over-specified linear system of $p$-adic equations

(Note: this is cross-posted with https://math.stackexchange.com/questions/4747805/min-norm-solution-of-over-specified-linear-system-of-p-adic-equations. I am posting it here as I got no answer after 1 ...
Andre's user avatar
  • 1
2 votes
0 answers
184 views

Multivariable Weierstrass preparation theorem

The Weierstrass preparation theorem for formal power series says the following: Let $f(T) \in \mathbf{Z}_p [[ T ]]$ be a formal power series. Then we can write $f(T) = p^{\mu} \cdot u(T) \cdot g(T)$, ...
Adithya Chakravarthy's user avatar
1 vote
1 answer
318 views

Unramified extension over $ \mathbb{Q}_{p} $

Let $\mathbb{Q}_{p}$ be a p-adic field such that $ p \neq 2 $. We knew that for every $ n=2m $ there exists exactly one unramified extension $ K $ of $ \mathbb{Q}_{p} $ of degree $ n $, obtained by ...
Sky's user avatar
  • 913
2 votes
1 answer
242 views

Twist of the Tate Curve

Suppose we have an elliptic curve $E$ over $K$, an $l$-adic field. Say that $|j(E)|>1$ where $|.|$ is the $l$-adic valuation. By the theory of the Tate curve $E$ is isomorphic over $L$ to a Tate ...
did's user avatar
  • 595
1 vote
1 answer
340 views

Does $17x^4+y^2=-1$ have solution in $\Bbb{Q}_2(\sqrt{-5})$?

This question raised when I tried to calculate $2$-Selmer group of elliptic curve $E:y^2=x^3+17x$ over $\Bbb{Q}(\sqrt{-5})$. $17x^4+y^2=-1$ does not have solution in $\Bbb{Q}_2$ (https://math....
Duality's user avatar
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3 votes
0 answers
507 views

Regarding the Challenge Problem in 3Blue1Brown's most recent video: Will $\binom{x}{4}+\binom{x}{2}+1=2^k$ for $x>10$? [duplicate]

Link to the video here with timestamp In deriving the formula for regions of Moser's Circle Problem, it observed that the formula $$ F(x)=\binom{x}{4}+\binom{x}{2}+1 $$ achieves values that are equal ...
wjmccann's user avatar
  • 315
7 votes
1 answer
320 views

Topological generators for $\mathrm{SL}_2(\mathbf{Z}_p)$

$\DeclareMathOperator\SL{SL}$ Let $p>3$ and $G$ be an open subgroup of the special linear group $\SL_2(\mathbf{Z}_p)$ over the ring $\mathbf{Z}_p$ of $p$-adic integers. Suppose that $G$ is ...
trivialquestions's user avatar
0 votes
0 answers
109 views

What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2 \mathbb{Z}_p$?

Let $p \geq 2$ be prime and $K=\mathbb{Q} (\zeta_p),~ \zeta^{p-1}=1$ with ring of integers $\mathcal{O}_K$. We denote $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ ...
MAS's user avatar
  • 870
2 votes
0 answers
52 views

Classification of submultiplicative ring norms on $\mathbb Q$

Let $R$ be a ring with identity. I call a non-negative real valued function $N: R \to \mathbb R_{\geq 0}$ a ring norm, if it has the following properties: $N(r) = 0$ iff $r = 0$ $N(r+s) \leq N(r) + N(...
Adelhart's user avatar
  • 227
12 votes
1 answer
540 views

$p$-adic L function of an odd Dirichlet character

Apologies for a naive question (especially for Iwasawa theorists): it is well-known and trivial to prove that the usual (elementary) construction of $p$-adic L functions attached to odd Dirichlet ...
Henri Cohen's user avatar
  • 11.9k
4 votes
0 answers
164 views

Does $p$-adic Baker theorem holds in the given case?

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, and $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$. Let $U_p$ be the units $(1+\mathfrak{m})$ of $\...
MAS's user avatar
  • 870
3 votes
0 answers
133 views

A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
Asvin's user avatar
  • 7,686
0 votes
1 answer
378 views

How do I extend the $2$-adic absolute value to prove Monsky's Theorem?

In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user avatar
3 votes
1 answer
408 views

$\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$?

$\newcommand\tetra[2]{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2), it has been pointed out how my ...
Marco Ripà's user avatar
  • 1,129
2 votes
1 answer
302 views

$p$-adic analogue of modular forms, upper half-plane, and $L$-functions

In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently ...
chbe's user avatar
  • 91
1 vote
1 answer
247 views

$p$-adic $L$-functions and congruence of $L$-values

I am reading about $p$-adic $L$-functions and I have one question in mind. To start with, I will write a proof I've learned of a congruence of $L$-values: Theorem: Let $p\geq5$ be a prime, $\alpha\...
ShBh's user avatar
  • 271
3 votes
0 answers
239 views

Nygaard filtration on Fontaine's period ring

Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. ...
user145752's user avatar
4 votes
1 answer
205 views

Existence of intermediate field extensions for tamely ramified p-adic extensions

Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
Ralph Morrison's user avatar
0 votes
1 answer
92 views

Sums of powers of measures of $p$-adic balls

Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...
Daniel Loughran's user avatar
5 votes
1 answer
235 views

Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$

To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...
a196884's user avatar
  • 323
3 votes
0 answers
111 views

Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
Piotr Pstrągowski's user avatar
2 votes
0 answers
148 views

Can there exist different smooth, proper schemes over the p-adics with the same generic fiber? [duplicate]

Can there exist smooth, proper $X_1,X_2/\mathbb Z_p$ such that their generic fibers are isomorphic but their reductions mod $p$ are not? Are there examples if we insist that the special fibers are ...
Asvin's user avatar
  • 7,686
2 votes
1 answer
188 views

Image of Kummer map for CM Elliptic curves

Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^*$ in $K$. ...
Adithya Chakravarthy's user avatar
7 votes
1 answer
420 views

Non-existence of "higher" Artin map

So rank $1$ local Langlands is special in as that it is given by the Artin map $$\text{GL}_1(K)\to G_K^{ab},$$ whereas in the higher rank (to the best of my knowledge) there doesn't exist a map $$\...
curious math guy's user avatar
0 votes
0 answers
119 views

How extension $\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ looks like?

Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$. I want to know what $\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ is . At first I tried to prove ...
Duality's user avatar
  • 1,415
2 votes
0 answers
176 views

Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])=\Bbb{Q}_p(\hat{E}[p])$

Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$. I want to prove $\Bbb{Q}_p(E[p])=\Bbb{Q}_p(\hat{E}[p])$. $ \hat{E}[p]$ denotes $p$ ...
Duality's user avatar
  • 1,415
2 votes
0 answers
184 views

How to plot a p-adic function? [closed]

I found on the Internet some ways to provide a graphical representation of the $p$-adic integers or numbers (e.g., these illustrations of Heiko Knospe). They all exploit the fact that $p$-adic ...
Perry's user avatar
  • 21
2 votes
0 answers
138 views

Bruhat-Tits tree as Cayley graph of free group

$\DeclareMathOperator\BT{BT}\DeclareMathOperator\GL{GL}$Let $p > 2$ be a prime and $n = \frac{p + 1}{2}$. We can identify the vertices of Bruhat-Tits tree $\BT(\mathbb Q_p)$ with the elements in ...
fyo's user avatar
  • 71
5 votes
0 answers
168 views

defining the upper ramification numbering

Given a local field $K$ with absolute Galois group $\Gamma$. Is it "possible" to define the upper numbering on $\Gamma$ without using the lower numbering? In other words, given $\gamma \in \...
Mark OSS's user avatar
  • 159

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