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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19
votes
1answer
602 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) ...
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0answers
161 views

Do algebraic completion/topological completion of fields always terminate? If so, are they unique?

Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$. On the other hand, the ...
5
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1answer
187 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 ...
8
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1answer
317 views

p-adic versions of log concavity for graphs (or matroids)

It was recently shown using techniques inspired by algebraic geometry (by Huh and Adiprasito-Huh-Katz) that the chromatic polynomial of a graph (or matroid) has coefficients that satisfy log-concavity....
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3answers
188 views

Is $K^\times/ F^\times$ compact for local fields?

Let $K/F$ be a finite extension of local fields (of characteristic 0). Is it true that the quotient group $K^\times/ F^\times$ is always compact? I understand that if the extension is cyclic, it is ...
2
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2answers
165 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 ...
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0answers
57 views

Continous morphisms of a local field with conditions in positive characteristic

Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...
6
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1answer
114 views

Hasse invariant and the Clifford algbera

Let $$q = a_1 x_1^2 + \cdots + a_n x_n^2$$ be a quadratic form over some $p$-adic field $\mathbb{Q}_p$. We thus have its Hasse invariant $$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...
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181 views

How to prove the map of rings $\mathcal{R} \to \mathcal{R'}$ is flat?

We fix a finite extension $K$ of $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $\kappa$. Consider the ring of witt vectors $W(\kappa)$ over the residue field $\...
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0answers
447 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 ...
6
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1answer
340 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:...
3
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1answer
73 views

An averaging procedure on finite multisets of $2$-adic integers

Recently there was this question talking about an averaging procedure on finite multisets of integers. After seeing that question, I thought about the same procedure but with integers replaced by $2$-...
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0answers
184 views

Is this topology on $\mathbb{Q}$ well studied?

Let $\|\cdot\|_p$ denote the $p$-adic norm on $\mathbb{Q}$. For the whole set of primes $P$ consider the topology which is generated with prebase of open sets $V_{p,\varepsilon}(x) = \{y\in\mathbb{Q} :...
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0answers
74 views

What would be the quotient groups $U_{\mathrm{gen}}/U_{\mathrm{gen}}^{(n)}$ and $U_{\mathrm{gen}}^{(n)}/U_{\mathrm{gen}}^{(n+1)}$?

Let $K \supseteq \mathbb{Q}_p$ be a $p$-adic field with ring of integer $O$ and maximal ideal $m$. Let $O^*$ be the group of units in $O$. Consider the group of units $U^{(0)}=U=O^*$ and $U^{(n)}=1+m^...
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0answers
78 views

How to define Weierstrass degree of higher dimensional $p$-adic power series?

Let $K \supset \mathbb{Q}_p$ be the $p$-adic field with ring of integer $O_K$ and unique maximal ideal $m_K$. If $f(x)$ be a power series in $O_K[[x]]$, then the Weierstrass degree of $f(x)$ denoted ...
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1answer
80 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 $(Q,V)$ ...
3
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1answer
113 views

Analytic p-adic functions that take an algebraic value

Suppose it exists $r\in\mathbb R$ such that the non constant p-adic function $f(z)=\sum_{n\ge0}a_nz^n$ ($a_n\in\mathbb C_p$) is defined on $\mathcal D=\{z\in\mathbb C_p\mid v_p(z)>r\}$. Does it ...
2
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0answers
58 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 ...
6
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2answers
564 views

Zero of the exponential p-adic

Consider the $p$-adic exponential defined over $\mathbb C_p$. One knows $\exp$ is analytic in the domain $\mathcal D=\{z\in\mathbb C_P\mid v_p(z)>\frac1{p-1}\}$. Does it exist an element $z_0\in\...
3
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0answers
104 views

Composition in function fields

Let $k=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. One has the map: $\circ:k\times\{v\in k\mid\deg(v)>0\}\to k$ defined by $f\circ g=\sum_{n\ge-m}a_ng^{-n}$ where $f=\sum_{n\ge-m}a_n\frac1{T^...
4
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1answer
252 views

Identity theorem in $p$-adic geometry/analysis

If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $\mathbb{Q}_p$ is disconnected. The modern approach to this is,...
6
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1answer
236 views

Perfect $\mathbb Z_\ell$-modules

Disclaimer : I asked this question on Maths.StackExchange 20 days ago (and started a bounty) here but got no answer, so I'm asking it here now (with no modification). $\newcommand{\l}{\ell} \...
3
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0answers
186 views

Definition of Iwahori subgroup independently of the Bruhat-Tits building

Let $G$ be the points of a connected, semisimple algebraic group over a $p$-adic field $k$. To make life easy, let's assume the underlying group scheme is simply connected. The Bruhat-Tits building $...
2
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0answers
106 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 ...
2
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1answer
276 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 ...
2
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0answers
200 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 ...
6
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0answers
200 views

intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$. I have some intuition for $\mathbb{Z}$-lattices ...
5
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0answers
260 views

Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics

Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-...
1
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0answers
62 views

valuation of a derivative in a completion

Let $q$ be a power of a prime $p$ and $w$ be an irreducible polynomial of $\mathbb F_q[T]$. Denote by $\mathbb C_w$ the completion of an algebraic closure of $K_w$, the completion of $\mathbb F_q(T)$ ...
3
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0answers
133 views

Jacobian change of variables formula for $p$-adic valued integration?

Let $k$ be a $p$-adic field. It's possible to make sense of the Haar measure $\mu_{\operatorname{Haar}}$ on $k^n$ as a $k$-valued measure and define integrals $$\int\limits_{k^n} f(x_1, ... , x_n) d\...
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0answers
105 views

continuous isomorphism of $p$-adic field

In Cassels-Frolich, one can read this theorem (page 57): Let $K$ be a finite finite separable extension of the valued field $(k,v)$. Let $\overline k$ be the completion of $k$ and $(K_j)_{1\le j\le r}$...
4
votes
2answers
285 views

$p$-adic series bounded if and only if it has finitely many zeros

Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\...
6
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0answers
78 views

On the isomorphism of the lattices of submodules of certain free modules

Let $K,L$ be two finite extensions of the $p$-adic field $\mathbb{Q}_p$ of the same degree. Let $\mathcal{O}_K$ and $\mathcal{O}_L$ be the ring of integers of these two fields, and let $\mathcal{O}_K^...
2
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1answer
118 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 $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ ...
2
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1answer
113 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 ...
3
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0answers
134 views

Using the Hilbert symbol to find nice field extensions

Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
0
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1answer
437 views

p-adic expansion of roots of unity [closed]

Let $w$ be an n-th root of unity, I have two questions 1) What are the conditions on the prime $p$ such that $w\in \mathbb{Z}_p$, and if it is the case what is the p-adic expansion of an n-th root of ...
0
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1answer
195 views

If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively prime modulo p, then do they generate $\mathbb{Z}_p[X]$?

I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample ...
4
votes
0answers
238 views

Computing the $2$-adic volume of a special orthogonal group

Let $n \geq 0$ be an integer, let $A = (a_{ij})$ be the $(2n+1) \times (2n+1)$ matrix defined by $a_{ij} = 0$ unless $i + j = 2n+2$, in which case $a_{ij} = 1$. Let $G$ be the group scheme over $\...
4
votes
0answers
111 views

Finite dimensional irreps of $p$-adic groups

What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$? One knows such a representations cannot be smooth, so probably the examples will be ...
6
votes
1answer
523 views

Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
1
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0answers
168 views

Extension of $\mathbb C_p$?

Let $K/\mathbb Q$ be a finite algebraic extension. Consider a prime $p$ and $v$ be a valuation of $K$ above the valuation $v_p$ of $\mathbb Q_p$. Denote by $K_v$ a completion for the valuation $v$ of $...
1
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0answers
63 views

extension of absolute values in function fields and product formula

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$. Let $K/\mathbb F_q(T)$ be a algebraic extension of finite dimension $...
2
votes
0answers
71 views

Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...
11
votes
2answers
591 views

Is there a proof of quadratic reciprocity using $p$-adic numbers?

I asked same question on MSE before, but I didn't get any answer yet. I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can ...
3
votes
0answers
111 views

Simple transfinite generalization of $p$-adic integers

One way to define the ring of $p$-adic integers is as a quotient of the formal power series semiring $\Bbb N[[x]]/(x-p)$. One can likewise start with the formal power series ring $\Bbb Z[[x]]/(x-p)$ ...
3
votes
1answer
285 views

Analytic continuation of a $p$-adic function

Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences of $\mathbb Q_p$ such that the function $f:z\in\mathbb Q_p\to\sum_{n\ge0}a_nz^n+b_nz^{n+1}$converges in $\{|z|_p<1\}$. Assume ...
1
vote
0answers
130 views

On $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le p-1}$ and $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le (p-1)/2}$

QUESTION. Is my following conjecture true? If true, how to prove it? Conjecture. Let $p$ be a prime with $p\equiv5\pmod 6$, and define the matrices $A_p$ and $B_p$ by $$A_p:=\left[\frac1{i^2-ij+j^2}\...
7
votes
1answer
674 views

Convergence of a $p$-adic series

Let $K$ be a local field of characteristic $0$ with valuation $v$. I think $$\lim_{\substack{q\in K\\q\to1}}\sum_{n\ge0}\prod_{j=1}^n\frac{q^j-1}{q-1}$$ converges to $\sum_{n\ge0}n!\in K$ but I did ...
4
votes
0answers
292 views

Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$

Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...