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

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11
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141 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
7
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80 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
5
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36 views

A p-adic integral

Let $(K,||)$ be a finite extension of $\mathbb{Q}_p$ of degree $d$ such that the restriction of $||$ to $\mathbb{Q}_p$ is the usual p-adic absolute value. Endow $GL_n(K)$ with the unique Haar measure ...
5
votes
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72 views

Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
5
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160 views

Is there a notion of *p-adic Dedekind Domains*?

As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$. Now is there any generalization such as the p-adic completions of a Dedekind Domain? This might be ...
4
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46 views

Question about topological properties of $\Bbb{C}_p$

It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) ...
4
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47 views

Group structure of $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$

Let p be 1 mod 3 (separate question: work out 2 mod 3). What is the group structure of the abelian group $\mathbb{Q}_p ^* / \mathbb{Q}_p ^{*3}$? $\mathbb{Q}_p ^*$ refers to the group of units in ...
4
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106 views

Why is the Euclidean metric called the prime at infinity?

I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
4
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80 views

If $(x_n)_{n\geq 1}$, $x_n \in \mathbb{Q}$ is p-adic Cauchy, show ord$_p(x_n)$ eventually constant.

Number Theory 1: Fermat's Dream asks the reader to verify the following. They then use this to extend the definition of Ord$_p$ to $\mathbb{Q}_p$. Let $p$ be prime and $a\neq 0$. If ...
4
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132 views

Liftings in unramified extensions of $\Bbb Z_p$

[Edit : I have changed the formulation of the question. Sorry for the trouble] Here is a stupid question, maybe trivial. Let $p$ be a prime number, $q = p^n$ where $n$ is an integer, $R = ...
4
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257 views

Intersection of nested open sets

Consider a nested sequence $O_1$, $O_2$, $\dots , O_k, \dots $ of open sets in $\mathbb{Q}_p^n$ such that $O_i \setminus O_{i+1}$ has empty interior. Under which conditions do we have $\bigcap_k O_k$ ...
4
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301 views

Ultrametric inequality

I am having trouble seeing the following consequence of the ultrametric inequality, which is supposed to be immediate. If $|x+y|\leq \max{\{|x|,|y|\}}$, then, equality holds when $|x|\neq |y|$. I ...
4
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367 views

p-adic numbers and binomial coefficients

Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ $${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$ This is again a ...
3
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128 views
+150

Every Cauchy sequence converges

SENTENCE: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. PROOF: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show ...
3
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43 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
3
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22 views

properties of certain semigroup action on $\mathbb{Z}/p\mathbb{Z}$

Suppose we have a polynomial $f \in \mathbb{Z}/p\mathbb{Z}[x]$, $f(x) = x^2 - x$. We are interested in elements $n \in \mathbb{Z}/p\mathbb{Z}$ such that after repeated application of f they ...
3
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29 views

GL$_2(\mathbb{Q}) Z_{\mathbb{R}}$ closed in GL$_2(\mathbb{A})$?

I am struggling with the following subgroups of GL$_2(\mathbb{A})$ where $\mathbb{A}$ is (the topological ring of) Adeles over $\mathbb{Q}$: $$G_\mathbb{Q} := \iota(\text{GL}_2(\mathbb{Q})) $$ where ...
3
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86 views

An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
3
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96 views

Constructing the complex p-adic numbers

I'm reading through "$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions" by Koblitz to learn about p-adic numbers. In chapter 3, he describes the construction of $\Omega$ (a.k.a. $\Omega_p$), the ...
3
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52 views

$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
3
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38 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
3
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46 views

computation of an $H^2$

Let $p$ be a prime number and $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $G_p$ be the absolute Galois group of $\mathbb{Q}_p$ and fix an absolutely irreducible representation $$ \rho : G_p \to ...
3
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61 views

Maximal compact subgroup of $GL_n(\mathbb C_p)$

It is known that the general linear group $GL_n(\mathbb Q_p)$ over the $p$-adic numbers has $GL_n(\mathbb Z_p)$ as a maximal compact subgroup and every other maximal compact subgroup of $GL_n(\mathbb ...
3
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52 views

Spectrum theorem for p-adic matrix analysis

Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\ 2: ...
3
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86 views

p-adic liftings on SAGE

I asked a question the other day: Multidimensional Hensel lifting which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the "algorithm" ...
3
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60 views

Green’s formula in p-adic integration

Is there an analogue of Green's formula in p-adic integration (with respect to the Haar measure)?
2
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14 views

no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal containing $p$ is a perfect $\mathbb{F}_p$-algebra?

I am reading the Notes on $p$-adic Hodge theory of O. Brinon & B. Conrad . Can someone explains the following things to me? «... no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal ...
2
votes
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40 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
2
votes
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28 views

$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
2
votes
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51 views

Equivalent Hensel's lemma?

Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$. I have seen two statements that do ...
2
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43 views

presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
2
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70 views

$p$-adic Integration

I am struggling to compute the following integral \begin{equation} \int_{\mathbb{Z}_p}\exp_p(|x|_p)d\mu \end{equation} where \begin{equation} \exp_p(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} \text{ defined ...
2
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72 views

Structure of $\mathbb{Q}_p(\zeta_p)$

Let $p \ne 2$ be prime number and denote by $\zeta_p$ the p-th root of unity. It's well known that $K = \mathbb{Q}_p(\zeta_p)$ has $t=1 - \zeta$ as prime element (generator of the Ideal $P_K = \{ x\in ...
2
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144 views

Can this function extend to a measure?

I believe I have one final question on a function I've been thinking about. To set up, let $\mathbb{Q}_p$ be the $p$-adic numbers, and let $B(x,r)$ denote the closed balls $$ ...
1
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37 views

Understanding the mechanics of P-adic topologies

I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open ...
1
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42 views

What does $p\mathbb{Z}_p$ mean?

I am looking at Hensel's Lemma: Let $F(x)=a_0+a_1x+ \dots + a_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p-$adic number $\alpha_1 \in \mathbb{Z}$, such that: $$F(\alpha_1) \equiv 0 ...
1
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0answers
18 views

Prove identities-p norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
1
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13 views

P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
1
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0answers
74 views

Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
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19 views

Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this ...
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55 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...
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65 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
1
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79 views

Open Ball under the p-adic Norm

I'm trying to figure how, if it's even possible, to draw an open ball using the p-adic norm. My definition of the p-adic norm I'm using is: $ \lvert x \rvert_p $ = $p^{-ord_px}$ if $x \neq 0$ and ...
1
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0answers
49 views

Question about $p$-adic exponential

Let $p$ be a prime number, and $K$ a finite extension of $\mathbb Q_p$ and $S=p^N\mathcal O_K$ where $\mathcal O_K$ the ring of integers of $K.$ I know that for $N>>0$ enough large the $p$-adic ...
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53 views

Topological isomorphism of projective system

Let $K_0 \subset K_1 \subset ...K_n\subset...$ be a tower of normal number fields, and put $G_n = \mathrm {Gal} (K_n/K_0).$ Define epimorphisms $\pi_{mn} :G_n\longrightarrow G_{m}$ for $m < n$ ...
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41 views

What do they mean by this: “the definition of product topo shows that this mapping is continuous”

Let $\Bbb{Z}_2$ be the $2$-adic integers. There's a bijection $\psi : \Bbb{Z}_2 \to C$, the Cantor set, defined by $\psi (\sum_{i \geq 0} a_i p^i) = \sum_{i \geq 0} \dfrac{2a_i}{3^{i+1}}$. The text ...
1
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0answers
62 views

Arithmetic in p-adic

I came to the problem saying that there exists a number $x\in \Bbb Z_7$ such that $x^2=2$ but there is no such $x$ in $\Bbb Z_5$. Could anyone give an explanation of this? How to actually find the ...
1
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0answers
84 views

Hensel's lemma in $\mathbb Z_2$

Can you give me a concrete example for a quadratic form $$ f(x,y)=ax^2+bxy+cy^2 \in \mathbb Z_2[x,y] $$ which has a primitive solution $(x^*,y^*) \in \mathbb Z_2 \times \mathbb Z_2$ (mod 4) with the ...
1
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0answers
70 views

Basic question about $p$-adic expansions

I was recently introduced to the $p$-adic numbers, and have been asked to show that for $n > 0$, the $p$-adic expansion of $\frac{1}{1 - p^n}$ is $\sum_{i=0}^\infty p^{in}$. Could someone tell me ...
1
vote
0answers
59 views

arithmetic of p-adics

I need only a fast check if the following expressions are correct. Let $\frac{a}{b} \in \mathbb{Q}\def\ord{\operatorname{ord}}$ then the following fact olds $|x|_p = \left|\dfrac{a}{b}\right|_p = ...