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
253 questions
3
votes
0
answers
198
views
Characterizing rational functions on $\mathbb{Q}$ in terms of smooth extensions to $\mathbb{R}$ and $\mathbb{Q}_p$
Consider a function $f$ from a cofinite subset of $\mathbb{Q}$ to $\mathbb{Q}$. As established here and here $f$ extending to a smooth function on a cofinite subset of $\mathbb{R}$ is not sufficient ...
- 13.2k
3
votes
0
answers
176
views
Carlitz factorials and Euler-like series
Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+...
- 3,998
3
votes
0
answers
412
views
Is $1+T$ a topological generator for $Z_{p}[[T]]$? [closed]
Consider the ring of formal power series $\mathbb{Z}_p[[T]]$ (where $\mathbb{Z}_p$ denotes the ring of $p$-adic integers) with the topology in which a neighborhood basis for $0$ is given by the ideals ...
- 147
3
votes
0
answers
218
views
Weil index computation, p-adic integral
The following peculiar p-adic integrals have arisen in my work, and I would be interested if anyone can see how to tackle them.
Let $F$ be a $p$-adic field, $\mathfrak{o}$ its ring of integers, $\...
- 123
3
votes
0
answers
246
views
Do $r$-th root Harmonic numbers ever sum to integers?
None of the Harmonic numbers $H_n = \sum_{k=1}^n 1/k$ are integers for $n>1$
(e.g., this MSE question and answer).
Q. Define the $r$-th root Harmonic number $H_n^{1/r} = \sum_{k=1}^n 1/{k^{1/r}}$...
- 151k
3
votes
0
answers
949
views
Transcendental numbers in the p-adic rationals $\mathbb Q_p$ [closed]
I know that there are uncountably infinite transcendentals over $\mathbb Q$ in $\mathbb Q_p$. What i want to ask is if there is a way to determine whether a transcendental over $\mathbb Q$ lays in ...
- 141
2
votes
1
answer
695
views
The Unit Group of $\mathbb{Z}_p$
Let $\mathbb{Z}_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}_p ^\times$ can be set theoretically described as $\bigcup _{1 \le a \le p-1} a+ p\mathbb{Z}_p$...
- 5,966
2
votes
2
answers
2k
views
A multidimensional version of Hensel's lemma? (for more than one polynomial)
The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy
$$
|f(a)|_p < | f'(a) |_p^2.
$$
Then there is a unique $\alpha \in \mathbb{Z}_p$...
- 3,625
2
votes
1
answer
640
views
Rings that inject in all p-adic integers
Denote $p$ a prime number and $\mathbb Z _p$ the ring of $p$-adic integers.
We have a canonical injective ring homomorphism $:\mathbb Z \rightarrow \mathbb Z_p$ for all $p$. But $\mathbb Z$ is not the ...
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
2
votes
1
answer
669
views
Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.
If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when ...
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
2
answers
546
views
A p-adic logarithm as a limit of discrete logs
I've been searching for something similar to the argument below for about a week now and I just must be missing out on the right key words. Can someone point me in the right direction and/or let me ...
- 2,235
2
votes
1
answer
499
views
On an isomorphism between $p$-adic power series and an inverse limit
Let $K$ be an extension field of $\mathbb{Q}_p$, let $O$ be the ring of integers of $K$, and let $P$ be the maximal ideal of $O$.
If $K$ is a finite extension of $\mathbb{Q}_p$, there is the well-...
- 3,107
2
votes
1
answer
256
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 ...
- 637
2
votes
2
answers
230
views
The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?
Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...
- 63
2
votes
1
answer
312
views
Why $E_1(\mathbb{Q}_p)\cong\mathbb{Z}_p$
I read an article where it is said: $E_1(\mathbb{Q}_p)\cong \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}\}$.
The ...
- 131
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
2
votes
1
answer
413
views
$2$-adic valuation on $\mathbb Q (\sqrt{-15})$
I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that
$-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ ...
- 243
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
2
votes
1
answer
263
views
Calculation of Tate epsilon factor in the ramified case
Let $F$ be a nonarchimedean local field, $\chi$ a ramified character of $F^{\ast}$, $\psi$ a nontrivial character of $F$, and $dx$ a Haar measure on $F$ with respect to which the Fourier transform is ...
- 6,180
2
votes
1
answer
442
views
Partitioning a compact open set into balls in an ultrametric space
Consider a $p$-adic field $K$ with the standard topology inherited from the usual $p$-adic norm $\mid \cdot \mid$. Consider the ultrametric space $X=K^n$ with the topology inherited from the norm $\| \...
user16974
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
2
votes
1
answer
170
views
Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$
Let $F$ be the spherically complete extension of $\mathbb C_p$ and $(a_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ such that for all $r\in\mathbb R$, one has $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ ...
- 3,998
2
votes
1
answer
202
views
Decomposition of $\widehat{k^{\times}}$ occuring in local class field theory
Let $k$ be a finite extension of $\mathbb{Q}_p$ very often we use the isomorphism that $Gal(\overline{k}/k)^{ab} \simeq \hat{(k^{\times})}$ given by local class field theory.
My question would be do ...
- 385
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
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
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
2
votes
0
answers
152
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{\...
- 21
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
2
votes
0
answers
68
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(...
- 237
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
2
votes
1
answer
202
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 $K$. ...
- 1,949
2
votes
0
answers
190
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 $p$ ...
- 1,541
2
votes
0
answers
211
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 ...
- 21
2
votes
0
answers
172
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 ...
- 71
2
votes
0
answers
256
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
2
votes
0
answers
47
views
Extension of morphism in local fields of positive characterisic
Consider $\theta:\mathbb F_q(T)\mapsto\mathbb F_q(T)$ defined by $\theta(Q)=Q(T^q)$. It is a morphism of fields. Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$. Then, $\theta$ can be ...
- 3,998
2
votes
0
answers
266
views
A "multi-adic" absolute value / topology?
Let $S$ be a set of finitely many prime numbers. Then, define $\left|\cdot\right|_{S}:\mathbb{Q}\rightarrow\left[0,\infty\right)$ by: $$\left|x\right|_{S}\overset{\textrm{def}}{=}\prod_{p\in S}\left|x\...
- 1,284
2
votes
0
answers
478
views
Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
- 930
2
votes
0
answers
75
views
Zeroes of the Euler series
Consider a prime $p$. Let $f$ be the Euler series defined by $f(z)=\sum_{n\ge0}n!z^n\}$. It is defined and analytic over $\mathcal D=\{z\in\mathbb C_p\mid v_p(z)>-\frac1{p-1}\}$. I try to check if ...
- 3,998
2
votes
0
answers
108
views
Can an abelian group have multiple different actions of $\mathbb{Z}_p$?
This is perhaps a trivial question, but I've asked a few colleagues and they couldn't answer. For a given abelian group $M$, is it possible to have several different actions of the ring of $p$-adic ...
- 21
2
votes
0
answers
226
views
Where is the flaw in this argument with $p$-adic extensions?
I cannot find what I am missing in the following computation. Let $K=\mathbb{Q}_p(p^{1/{(p-1)p^{\infty}}})$ and $L=\mathbb{Q}_p(\zeta_{p^{\infty}})$, where $\zeta_{p^n}$ is a primitive $p^n$-th root ...
- 282
2
votes
0
answers
69
views
Invariant compact in division ring
Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...
- 3,998
2
votes
0
answers
102
views
Zero digits of a p-adic algebraic number
This question might be too simple and I just don't see something very obvious, so I apologize in advance if that is so.
Let $p$ be a prime number and let $\mathbb Z_p$ denote the ring of $p$-adic ...
- 1,625
2
votes
0
answers
140
views
Completing vs Extending a field
Given a field and a metric on it, consider the goal of completing it and extending it in order to get an algebraicly closed and complete field.
How should one proceed? Should one first complete it ...
- 1,638
2
votes
0
answers
166
views
Relative Leopoldt defect
Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...
- 1,066
2
votes
0
answers
506
views
p-adic Lie theory
It is well known that exponential map in $C^{n\times n}$ will cover all non-sigular matrix $GL(n,C)$, which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases.
...
- 1,706
1
vote
2
answers
1k
views
Some good journals in $p$-adic number theory
Over the past few decades, a vast research area in number theory is surrounded by the $p$-adic field $\mathbb{Q}_p$ and its extensions.
My question is on different perspective.
What are the lists of ...
- 930
1
vote
1
answer
885
views
Etale cohomology in the $p$-adic setting
Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$?
Recall that semialgebraic subsets are obtained from $p$-adic ...
user16974