All Questions
Tagged with p-adic-numbers nt.number-theory
105 questions
4
votes
0
answers
91
views
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
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
2
votes
1
answer
250
views
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 \...
anon
0
votes
0
answers
81
views
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
views
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 ...
- 367
3
votes
0
answers
115
views
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
views
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 ...
- 31
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 ...
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 ...
- 549
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)$...
- 785
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 $\...
- 63
1
vote
1
answer
253
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}...
- 15.6k
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 ...
- 423
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....
- 1,541
3
votes
0
answers
511
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 ...
- 315
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 ...
- 13.1k
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
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 ...
- 1,451
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 ...
- 91
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 $L$ (...
- 421
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 ...
- 21.4k
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 ...
- 323
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
$$\...
- 4,428
5
votes
0
answers
181
views
defining the upper ramification numbering
Given a local field $K$ with absolute Galois group $\Gamma$. Is it "possible" to define the upper numbering on $\Gamma$ without using the lower numbering?
In other words, given $\gamma \in \...
- 159
5
votes
1
answer
190
views
Describing the Gamma-transform explicitly in terms of power series
The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma_{\alpha}(s) = \int_{\mathbf{...
- 1,949
10
votes
0
answers
248
views
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 ...
- 101
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 ...
- 707
6
votes
0
answers
228
views
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
257
views
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
views
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)...
- 163
3
votes
1
answer
529
views
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 ...
- 3,998
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
views
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
views
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
$$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$
...
- 17.9k
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. ...
- 7,019
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
views
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 ...
user178596
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 $$...
user178596
23
votes
1
answer
3k
views
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) ...
5
votes
1
answer
453
views
Computing the $p$-adic gamma function $\Gamma_p$
Let $p>2$ be a prime. For $n \in \mathbb{Z}^+$ we can define
\begin{equation}
F(n) = (-1)^n \prod_{1<i<n, p\nmid i} i.
\end{equation}
Since $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, we can ...
- 707