All Questions
Tagged with p-adic-numbers nt.number-theory
105 questions
4
votes
0
answers
91
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Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
- 6,622
4
votes
1
answer
239
views
A specific $2$-dimensional Galois representation of $G_{\mathbb{Q}_2}$ and its Langlands correspondence
I am interested in understanding a situation in (classical, not $p$-adic) local Langlands for $\mathrm{GL}_p(\mathbb{Q}_p)$. An example of
it is as follows: Let $F=\mathbb{Q}_2$ and $E$ be the ...
- 1,856
2
votes
1
answer
250
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A question on distinguished pairs
I am reading Alexandru, Popescu, and Zaharescu, "On the Closed Subfields of $\mathbb{C}_p$" (see https://tinyurl.com/kknmzbyx). The authors give the following definition:
Let $\alpha, \beta \...
CommunityBot
- 1
2
votes
0
answers
110
views
Galois action on the cohomology of a curve over a local field with bad reduction
Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\...
- 7,746
0
votes
0
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81
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What is the action of the Galois group due to Lubin-Tate formal group on the respective Tate module?
It is a well-known fact that a Tate module $T_p(A)$ of an abelian group (abelian variety or commutative group scheme) $A$ over a field $K$, equipped with a continuous action of the respective absolute ...
- 195
2
votes
1
answer
179
views
Ramification at particular level of a tower of extensions of local field
Let $K$ be an unramified extension of the $p$-adic number field $\mathbb{Q}_p$.
Suppose we have a tower of extensions:
$$K=:K(u_0) \subset K(u_1) \subset K(u_2) \subset K(u_3) \subset \cdots \subset ...
- 195
0
votes
1
answer
170
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Integral closure in the algebraic closure of $p$-adic numbers
Let $p$ be a prime number and let $\overline{\mathbb{Q}}_p$ be a fixed algebraic closure of the $p$-adic numbers $\mathbb{Q}_p$. It is well know that the ring of integers of $\mathbb{Q}_p$ is the ring ...
- 875
3
votes
0
answers
115
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Problem Deducing the value of Quadratic Hilbert Symbol from Explicit Formulas
This question concerns the explicit law for the Hilbert Symbol given in Sur les lois de réciprocfites explicites I by Henniart. I am trying to deduce the classical value of the Hilbert Symbol in $\...
- 171
3
votes
1
answer
272
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Does there exist a polynomial that extracts the highest digit of an integer in base p?
Given an odd prime $p$, a positive integer $1 \lt n$, and an integer $x \in \mathbb{Z}/p^n\mathbb{Z}$, does there exist an an integer-coefficient polynomial that extracts the highest digit of $x$?
The ...
4
votes
1
answer
585
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 ...
- 492
2
votes
0
answers
92
views
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 ...
- 930
5
votes
0
answers
197
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 ...
3
votes
1
answer
180
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 ...
- 149k
4
votes
1
answer
367
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 ...
- 149k
2
votes
1
answer
150
views
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)$...
- 32.5k
1
vote
1
answer
150
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 $\...
- 12.9k
1
vote
1
answer
253
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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}...
- 528
2
votes
1
answer
223
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 ...
- 16.2k
2
votes
0
answers
257
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 $f(T) = p^{\mu} \cdot u(T) \cdot g(T)$, ...
- 1,949
1
vote
1
answer
381
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 ...
- 923
1
vote
1
answer
348
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....
- 10.8k
3
votes
0
answers
511
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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 ...
- 34.4k
12
votes
1
answer
579
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 ...
- 37k
3
votes
1
answer
437
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 ...
- 12.9k
12
votes
1
answer
841
views
Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$
Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...
CommunityBot
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4
votes
0
answers
170
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 $\...
- 930
3
votes
0
answers
145
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 ...
- 7,746
2
votes
1
answer
356
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 ...
- 12.9k
1
vote
1
answer
280
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\...
- 302
4
votes
1
answer
243
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$ (...
- 2,959
0
votes
1
answer
96
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 ...
- 63.9k
5
votes
1
answer
275
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 ...
- 20.4k
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 ...
- 7,746
7
votes
1
answer
424
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},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\...
- 37k
5
votes
0
answers
181
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 \...
- 159
5
votes
1
answer
190
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Describing the Gamma-transform explicitly in terms of power series
The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma_{\alpha}(s) = \int_{\mathbf{...
- 37k
2
votes
1
answer
174
views
Finding a certain value of $\Gamma_p$
Let $\Gamma_p : \mathbb{Z}_p \to \mathbb{Z}_p^{\times}$ be the $p$-adic gamma function. I thought that I had successfully calculated $\Gamma_p(1 - 1/4)$, but sage is telling me I'm wrong (this is ...
CommunityBot
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10
votes
0
answers
248
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Some variants of Artin's primitive root conjecture
Artin's primitive root conjecture asserts that if an integer $a \ne -1$ is not a perfect square then $a$ is a primitive root mod $p$ for infinitely many primes $p$. This conjecture is still open.
An ...
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0
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228
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Are the $p$-adic digits of roots of unity equidistributed?
I was looking at the $p$-adic expansions of roots of unity in $\mathbf{Z}_p$, and I noticed that the digits tended to be equidistributed among the numbers $\{0, 1, \dots, p-1 \}$. I wanted to ask if ...
- 1,949
2
votes
0
answers
256
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Is the absolute Galois group $\text{Gal}(\bar K/K)$ isomorphic to $\text{Gal}(K(S)/K)$?
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and uniformizer $\pi$. Let $\bar K$ be the algebraic closure of $K$ and $\bar{\...
- 930
10
votes
1
answer
512
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p-adic analogue of octonions
There are the complex p-adic numbers.
But what is the p-adic analogue of the Cayley–Dickson construction?
Or more important: What is the p-adic analogue of the octonions?
It would be nice if the (unit)...
- 32.5k
3
votes
1
answer
529
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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 ...
- 63.9k
0
votes
1
answer
607
views
Method to solve modular quadratic polynomial [duplicate]
If $q$ is a prime what is the best method to compute roots of a quadratic polynomial $f(x)\equiv0\bmod q^2$ which is of form $x^2+bx+c\equiv0\bmod q^2$ where $b^2-4c\equiv0\bmod q$ and $gcd(b,q)=1$ ...
- 13.9k
18
votes
1
answer
714
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Is the p-adic density of the image of a polynomial always rational?
This question was previously posted here on MSE.
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. For $n\in\mathbb N$, let $I_n$ be the number of integers $i\in\{1,\...
- 654
3
votes
0
answers
163
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Reconstructing elements of $\mathbb Q$ in $\mathbb Z_p$
Can a rational number $a/b$ (with $b$ coprime to a prime number $p$) be recovered efficiently from a $p$-adic expansion of the form
$$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$
...
- 63.9k
23
votes
1
answer
3k
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A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
CommunityBot
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7
votes
1
answer
881
views
A family of Diophantine equations with no integer solutions but solutions modulo every integer
Selmer's curve is the equation $3x^3 +4y^3 +5z^3=0$. This equation is famous for having non-trivial solutions in every completion of $\mathbb{Q}$ but only having the trivial solution in the rationals. ...
- 2,821
0
votes
0
answers
202
views
When is $u \circ v=v \circ u$ for $p$-adic power series $u$ and $v$ in two power series rings $A$ and $B$ respectively?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field with ring of integers $O_K$ and maximal ideal $m_K$. Let $\bar K$ be the algebraic closure and $\bar{m}_K$ be the integral closure of $m_K$ with ...
- 930
10
votes
3
answers
1k
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What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
- 63.9k
1
vote
0
answers
94
views
How to estimate the highest power of 2 in the partial sum of 2-adic $\log(-1)$ (i.e. $\sum_{i=1}^n\frac{2^i}{i}$)?
The estimate I wanna get is $$v_2(\sum_{i=1}^n\frac{2^i}{i})\geq\min_{t\geq n+1}\{t-v_2(t)\}\tag{*}$$
where $v_2$ is the 2-adic valuation, that is the highest power of 2 defined on $\mathbb{Q}$.
Set $$...
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