Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [p-adic-number-theory]

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.

3
votes
1answer
49 views

$p$-adic supremum of cyclotomic polynomial

Let $p$ be a prime number, and $\Phi_{n}(T)$ be the $p^{n}$th cyclotomic polynomial, which we consider as a function on $\mathbb{C}_{p}$. In Pollack's paper 'On the $p$-adic L-function of a modular ...
1
vote
1answer
38 views

Absolute convergence of Fourier series in $\mathbb{Z}_p$

For a $p$-adic number $a \in \frac{n}{p^k}+\mathbb{Z}_p\subset \mathbb{Q}_p$ let $\exp(2i \pi a) =\exp(2i \pi \frac{n}{p^k})$ and $\psi_a(x) = \exp(2i \pi ax)$. Then $$Hom(\mathbb{Z}_p,\mathbb{C}^\...
0
votes
0answers
31 views

Proof on a generalization of Hensel's lifting lemma

I am reading a proof on the generalization of Hensel's lifting lemma (over p-adic integers), particularly on 1-lipschitz functions. I came across a part with notations which I've never encountered ...
2
votes
1answer
79 views

Shw that $ \ (1+X)^a \in \mathbb{Q}$ both in $\mathbb{R}$ and $ \mathbb{Q}_p$ under some condition

$\underline{\text{p-adic numbers and p-adic power series}}:$ If $ a,X \in \mathbb{Q}$, then when the binomial expression $f(X)=(1+X)^a \in \mathbb{Q} \ $ both in p-adic field $\mathbb{Q}_p$ and real ...
1
vote
0answers
76 views

Proof of infinite limit

My question is really as to whether I can consider the following results as proof of one another, since $(2a)$&$(2b)$ cannot be true unless $(0)$ is true, and vice versa. So if for below I prove $...
1
vote
0answers
43 views

Analog of finite set in valued complete division ring

Let $K$ be a complete valued division ring and $S$ be a compact subset of $D$. It is easy to see that if $K$ is commuative, the $K$-algebra $\mathscr C(S,K)$ of continuous functions on $S$ in $K$ is $...
3
votes
1answer
29 views

Admissibility of representations

Consider a smooth representation $(\pi,V)$ of the group $G=GL_n(\mathbb{Q}_p)$. The representation is said to be admissible if for every open compact subgroup $K \subseteq G$, the space $V^K$ of $K$-...
3
votes
1answer
49 views

Ring of integers of $\mathbb{C}_p$

Sorry, maybe this is a really stupid question. Let $\mathbb{C}_p$ be the completion of the algebraic closure $\overline{\mathbb{Q}_p}$ of the field of $p$-adic numbers. We know that there exists a way ...
0
votes
0answers
34 views

Is there any chance that this power series converges p-adically in the rational fieldin $\mathbb{Q}$?

Consider the p-adic power series $ \sum _{n=1}^{\infty} \frac{x^n}{u^n}$. Consider the usual absolute value $|.|$ and p-adic absolute value $|.|_p$ on $ \mathbb{Q}$, where $ u \in \mathbb{Q}$. Is ...
4
votes
0answers
63 views

Is there any interesting description of $\Bbb Q_p^\times / \Bbb Q^\times$?

Trying to reply to a comment to this answer of mine, I realised I know no better description of the quotient of multiplicative groups $\Bbb Q_p^\times / \Bbb Q^\times$ than just that. Of course I am ...
4
votes
0answers
66 views

Differentiating a $p$-adic character

Let $L$ be a finite extension of $\mathbb Q_p$ with ring of integers $\mathcal{O}=\mathcal{O}_L$ and let $B_1(L):=\{z \in L \colon \vert z-1 \vert <1 \}$. Let $\widehat{\mathcal{O}}(L)_{\mathbb ...
1
vote
1answer
94 views

Is $\mathbb Z_3+\mathbb Z_3\mathbf i+\mathbb Z_3\mathbf j+\mathbb Z_3\mathbf k$ a local ring?

I consider the quaternion division ring on $\mathbb Q_3$: that is $$\mathbb H_{\mathbb Q_3}=\{a+b\mathbf i+c\mathbf j+d\mathbf k \mid a,b,c,d\in\mathbb Q_3\}$$ with $\mathbf i^2=\mathbf j^2=\mathbf k^...
3
votes
4answers
159 views

Does $\text{SO}_2(\mathbb{Q}_5)$ contain non-trivial elements?

I was trying to find an element of $\text{SO}_2(\mathbb{Q}_5)$ for the $5$-adic numbers. By analogy with $\text{SO}_2(\mathbb{R})$ $$ \left[ \begin{array}{rr} a & -b \\ b & a \end{array} \...
0
votes
2answers
33 views

Are there any equilateral triangles in a p-adic field?

I'm struggling to find an equilateral triangle in $\Bbb Z_2$ so I wondered if there aren't any. Suppose the triangle $\{x,y,z\}$ If this is to be equilateral then $\lvert x-y\rvert_p=\lvert z-y\...
0
votes
0answers
51 views

What's a geodesic in $\Bbb Q_2$?

What's a geodesic in $\Bbb Q_2$? I want to better understand the concept of p-adic geodesics, just at a fairly basic level. As for a little background and my own attempt, all I can do is make a wild ...
1
vote
0answers
30 views

How to compute the $p$-adic period of $\mathbb G_m$?

I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $\mathbb G_m$: I know the example for the ...
2
votes
0answers
39 views

Roadmap to rigid cohomology

I am interested in the study of p-adic geometry. Unfortunately what I know is basic Algebraic geometry and basic number theory. To get an idea of the amount of material to study what could be a "...
3
votes
1answer
56 views

Determine whether the following quadratic forms are isotropic over $\mathbb{Q}$.

$q(x, y, z) = 5x^2-y^2-11z^2$ $q(x, y, z) = 3x^2-y^2+22z^2$ So for part (1), I apply Hasse-Minkowski and check over the $\mathbb{Q}_p$. It's trivial to see for $\mathbb{R}$. We only need to check for ...
1
vote
0answers
91 views

$\operatorname{ord}_p(\sum_{i=0}^{p-1}T(\overline i)^{p-1-k}\psi(\overline i))=?$

Let $Z_p$ denote the p-adic integers. Let $T:\mathbb{F_p}\to Z_p$ be a function with the following properties: $\forall x \in \mathbb{F}_p[\overline {T(x)}=x]$ $\forall x \in \mathbb{F}_p[T(x)^p=T(x)]...
1
vote
0answers
24 views

what would be the solution by using Hensel's Lemma?, p-adic numbers

Point out the main difference or relation between Newton's polygon and Hensel's lemma when it comes to find solution of the two variable polynomial $ f(x,y)=y^6-5xy^5+x^3y^4-7x^2y^2+6x^3+x^4=0$. $ \...
5
votes
0answers
46 views

Ordered subfields of $\mathbb{Q}_p$

I recently read about real ordered fields. Using real closures, I figured out that for each algebraically closed field $C$ of characteristic $0$, there exists a real closed subfield $R\subseteq C$ ...
0
votes
1answer
33 views

Valuation of the p adic logarithm

I'm stuck in some propertie about the $p$-adic logarithm. The propertie comes from a proposition in a Book by Dwork which I'm studying. The proposition says: If $v_{p}(x)>\frac{1}{p-1}$, then $v_{...
1
vote
1answer
48 views

Proving $ord_p(ζ_p-1)=1/(p-1)$

After proving this, I was able to deduce an even more general result that $ord_p(ζ_p-1)=ord_p(ζ_p^2-1)=...=ord_p(ζ_p^{p-1}-1)$. Now, according to Lubin, $ord_p(ζ_p-1)$ should be $1/(p-1)$, but this ...
2
votes
3answers
43 views

Showing the lemma $\operatorname{ord}_p(1+ζ_p)=0$ if $p>2$

Just to give some background regarding my motivation, I'm trying to prove a lemma to help me solve How do we prove p-order of $g_k$ is $\frac {k} {p-1}$? Let $Z_p$ denote the p-adic integers, and let ...
0
votes
1answer
38 views

Proof that a finite series expansion of $f(X)$ at $\alpha$ exists iff $Q(X)$ is a power of $(X-\alpha)$, in $f(X)=\frac{P(X)}{Q(X)}$

I'm working through Gouvea's P-adic numbers book, and early on they give the problem Write $f(X)=\frac{P(X)}{Q(X)}$ in lowest terms, so that $P(X)$ and $Q(X)$ have no common zeros. Show that the ...
1
vote
1answer
34 views

Homomorphism from $p$-adic to $l$-adic groups

I have seen and heard the statement that the $p$-adic and $l$-adic topologies are incompatible. I would appreciate a proof or references supporting this statement. More precisely, I am interested in ...
2
votes
0answers
47 views

Subgroup generated by union of two maximal compact subgroups of $GL_2(\mathbb{Q}_p)$

Let us denote by $G:= GL_2(\mathbb{Q}_p), G_0:= GL_2(\mathbb{Z}_p), g:= \begin{bmatrix}0 & 1\\p& 0\end{bmatrix}$ and by $G_1:= g G_0 g^{-1}$. I want to know if we have a good description of ...
2
votes
0answers
27 views

Upper-triangular subgroup is not unimodular

Consider the group $GL_n(\mathbb{Q}_p)$ of $n \times n$ invertible matrices over the $p$-adic field $\mathbb{Q}_p$. My goal is to prove that the subgroup $P_0$ of upper triangular matrices is not ...
7
votes
1answer
120 views

an inverse of the Artin-Hasse exponential?

In the p-adic world the Artin-Hasse exponential is the sollowing power series: $$ E_p(x)= \exp \left( \sum_{n=0}^{\infty}\frac{x^{p^n}}{p^n} \right) $$ where $E_p(x)\in 1+x\mathbb{Z}_{(p)}[[x]]$ ...
0
votes
1answer
19 views

Principal series for $\operatorname{GL}_2$, question about an exact sequence

I have a question about Proposition 7.2 of these notes by Gordon Savin. Here $G = \operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, and $\...
3
votes
1answer
57 views

What is p-adic logarithmic map of an elliptic curve? How to compute it?

I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...
0
votes
1answer
29 views

books on p-adic cryptography

I'm currently studying this paper: ''Modular and p-adic Cyclic Codes'', A. Robert Calderbank, N. J. A. Sloane, published 1995 in Designs, Codes, Cryptography DOI:10.1007/BF01390768 but am having a ...
2
votes
1answer
44 views

Interpolation with $p$-adic formal power series in $\mathbb Z_p[[x]]$

It's a classical exercise in complex analysis that one could find a holomorphic function with given values on a set of points on the complex plane without limit points. What about the p-adic analogue?...
1
vote
1answer
36 views

Find the set of interstion of $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ shwoing its elements [duplicate]

This question is from $\text{p-adic numbers}.$ My questions are- $(1)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty? If non-empty, then what are the elements or ...
0
votes
1answer
55 views

Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty? [closed]

$\text{p-adic numbers}:$ My questions are- $(1)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty? $(2)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \...
1
vote
0answers
32 views

Results/topics on p-adic number theory analysis understandable by undergraduates

I've been reading about p-adic analysis lately, and am interested to work on a specific topic about it for my undergraduate thesis. However, almost all dissertations I've read online are too hard to ...
4
votes
1answer
62 views

p-th powers in p-adic field

Denote by $K$ the completion of $\bigcup_{n \geq 1} \mathbb{Q}_p (\zeta_{p^n})$, where $\zeta_{p^n}$ is a $p^n$-th root of unity. Is it true that any element in $K$ is a $p$-th power of some element ...
3
votes
1answer
47 views

Lower bound for the valuation of the trace of an element in a $p$-adic cyclotomic extension.

I'm reading Pierre Colmez's Fonctions d'une variable p-adique and he summons the following inequality without proof: Let $F_{n} := \mathbb{Q}_{p}(\zeta_{p^{n}})$ where $p$ is prime. Let $x \in F_{n}$,...
2
votes
2answers
80 views

How to find the Newton polygon of the polynomial product $ \ \prod_{i=1}^{p^2} (1-iX)$

How to find the Newton polygon of the polynomial product $ \ \prod_{i=1}^{p^2} (1-iX)$ ? Answer: Let $ \ f(X)=\prod_{i=1}^{p^2} (1-iX)=(1-X)(1-2X) \cdots (1-pX) \cdots (1-p^2X).$ If I multiply , ...
0
votes
0answers
21 views

I got stuch finding Newton polygon of the following product with any easiest method

Find the Newton polygon of the following polynomials: $(i) \ f(X)=(1-X)(1-pX)(1-p^3X)$, $ (ii) \ g(X)=\prod_{i=1}^{p^2} (1-iX)$. Answer: $(i)$ To find the Newton polygon for the polynomial $f(X)$,...
0
votes
0answers
29 views

Newton polygon : Show that precisely $ l$ of the $ \lambda_i$ are equal to $ \lambda$

$\text{Newton Polygons for Polynomials}$ There is a lemma in the book $ \ \text{p-adic numbers, p-adic analysis and zet-functions} $ of the author $ \text{Neal Koblitz} \ $ which I mentioned below: $...
1
vote
1answer
51 views

Uniqueness in Weierstraß p-adic preparation theorem

I have given the Weierstraß p-adic preparation theorem. It is stated as follows: Let $f=a_0 + a_1T + ... \in \mathbb{Z}_p[[T]]$ for a prime $p$ such that $p \mid a_0,...,a_{n-1}$ and $p\not\mid a_n$. ...
1
vote
1answer
46 views

Choice of a square root in $\mathbb{Q}_7$

While studying a calculation in Koblitz's P-adic Numbers, P-adic Analysis, and Zeta-Functions, he remarks the following: ...we were sloppy when we wrote $4/3 = (1 + 7/9)^{1/2}$. In both $\mathbb{R}...
5
votes
0answers
139 views

(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$

I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates). $\...
4
votes
0answers
29 views

Multiplicative Group of $p$ - adic Power Series

We let $D$ be an open or closed disc in $\mathbb{C}_p$ centered at $0$. I have to prove that $\text{(a)}$ $\{f \in 1 + x\mathbb{C}_p[[x]] : f \text{ converges in }D\}$ is closed with respect to ...
2
votes
0answers
31 views

Convergence of a $p$ - adic power series

We have a sequence $(a_n) \in \mathbb{C}_p$ such that $p^n||a_n|| \rightarrow 0$. I have to prove that the series $$\sum_{n=0}^{\infty}a_n \frac{n!}{x(x+1)(x+2)\cdots(x+n)}$$ converges $\forall x \in \...
0
votes
0answers
21 views

$\|a\| = \sup_{p \le \infty} |a|_p$ and prime number theorem

For $a \in \mathbb{Q}$, let $\|a\| = \displaystyle\sup_{v \le \infty} |a|_v$ where $|.|_\infty$ is the absolute value on $\mathbb{R}$ and $|.|_p$ is the absolute value on $\mathbb{Q}_p$ with the ...
0
votes
1answer
40 views

Any multiplicative character $χ: \mathbb{Z}/p\mathbb{Z} \to Z_p$ is a power of the Teichmuller character

This is a follow-up question to Existence/uniqueness of the Teichmuller map for p-adic integers Let $Z_p$ denote the p-adic integers. Let $π:Z_p \to \mathbb{Z}/p\mathbb{Z}$ by $π(a_0+a_1p+a_2p^2+...)=...
0
votes
2answers
51 views

Existence/uniqueness of the Teichmuller map for p-adic integers

Let $Z_p$ denote the p-adic integers. Let $π:Z_p \to \mathbb{Z}/p\mathbb{Z}$ by $π(a_0+a_1p+a_2p^2+...)=a_0$ where, of course, each $a_i \in \{0,1,...,p-1\}$ According to my professor, there exists ...
0
votes
0answers
30 views

Determining the Prime Order of a Prime Power Factorial

Let $\text{ord}_p{(n)}$ denote the largest power of $p$ that divides $n$. Thus for example, $$\text{ord}_5{(250)}=\text{ord}_5{\left(2\cdot 5^3\right)}=3$$ I want show that for a fixed $N$ and $p$,...