All Questions
Tagged with p-adic-numbers nt.number-theory
105 questions
0
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answers
202
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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
6
votes
1
answer
263
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
$$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...
- 103
2
votes
0
answers
478
views
Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
- 930
1
vote
0
answers
94
views
How to estimate the highest power of 2 in the partial sum of 2-adic $\log(-1)$ (i.e. $\sum_{i=1}^n\frac{2^i}{i}$)?
The estimate I wanna get is $$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
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
22
votes
1
answer
3k
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What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?
In the Boston conference on Fermat's Last Theorem (Summer 1995), Barry Mazur said (around 15m into the video) about class field theory that
If you are a number-theorist and you want to cheer ...
- 18.7k
32
votes
2
answers
3k
views
Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?
Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
"proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
- 7,347
5
votes
0
answers
328
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-...
- 461
11
votes
2
answers
813
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lowest degree of polynomial that removes the first digit of an integer in base p
Let $p$ be an odd prime and $n \geq 2$.
(1) Does there exist an integer-coefficient polynomial $f$ such that $f(x) = x - (x \bmod p)$ for all $x \in \mathbb{Z}/p^n \mathbb{Z}$? The polynomial ...
- 75
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
6
votes
1
answer
716
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Integral Tate-Sen theory
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...
- 23k
1
vote
0
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94
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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^...
- 930
14
votes
1
answer
1k
views
A quantitative version of Hensel's Lemma
I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts the number of lifts of a $\mathbb{Z}/p\mathbb{...
- 21.4k
7
votes
2
answers
606
views
convergence in $\hat{\mathbb{Z}}$, modulo prime power
The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16).
Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$
by $a_0=b, a_{n+1}=2^{a_n}$. Prove that $...
- 101
4
votes
2
answers
2k
views
Automorphisms of $\mathbb C_p$
I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...
- 3,998
5
votes
2
answers
528
views
2-adic valuation of odd harmonic sums
(This question is cross-posted on math.stackexchange)
I'm playing with p-adic valuations, and find that the odd harmonic sums, $\tilde{H}_k=\sum_{i=1}^{k}\frac{1}{2i-1}$, has 2-adic valuation $||k^2||...
- 355
4
votes
0
answers
442
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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(\...
- 685
3
votes
0
answers
164
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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 $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
- 2,516
25
votes
1
answer
833
views
Only finitely many values of the symmetric functions of $1/1,1/2,\ldots,1/n$ are $2$-adic integers (?)
For integers $n \geq k \geq 1$ let
$$H(n,k) := \sum_{1 \leq i_1 < \cdots < i_k \leq n} \frac1{i_1 \cdots i_k}$$
be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \...
user40023
4
votes
0
answers
124
views
Finite dimensional irreps of $p$-adic groups
What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$?
One knows such a representations cannot be smooth, so probably the examples will be ...
- 81
6
votes
3
answers
2k
views
structure of norm one group for quadratic extension of p-adic fields
Let $F$ be a p-adic field (finite extensions of $\mathbb{Q}_p$ for some prime $p$), and $E/F$ be a quadratic extension. Use $\sigma$ to denote the nontrivial element in the Galois group $Gal(E/F)$. ...
- 2,709
1
vote
0
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147
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On $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le p-1}$ and $\det\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le (p-1)/2}$
QUESTION. Is my following conjecture true? If true, how to prove it?
Conjecture. Let $p$ be a prime with $p\equiv5\pmod 6$, and define the matrices $A_p$ and $B_p$ by
$$A_p:=\left[\frac1{i^2-ij+j^2}\...
- 15.6k
5
votes
1
answer
670
views
Linear independence of p-adic logarithms (analog of Baker's theorem)
We have the following theorem of Baker:
Theorem 1. Let $\alpha_1, \ldots, \alpha_m \in \mathbb{C}$ be algebraic numbers $\neq 0, 1$ such that $\log \alpha_1, \ldots, \log \alpha_m$ are linearly ...
user40023
11
votes
0
answers
1k
views
Nick Katz observation: "the rationality of the zeta function!"
In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-...
- 3,107
6
votes
2
answers
2k
views
Trace of n-th root of unity in cyclotomic extension of p-adic rationals
Let $n\in\mathbb N$ and $p$ be any prime. Denote by $\mathbb Q_p$ the $p$-adic numbers. For a field extension $L/K$ denote by $Tr_{L/K}$ the corresponding trace function.
Let $\zeta_n$ be a primitve $...
- 115
5
votes
2
answers
934
views
Absolutely irreducible p-adic representation of the absolute Galois group of Q_p
Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, $G_p$ the absolute Galois group of $\mathbb{Q}_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}_p$. Assume we are ...
- 107
3
votes
1
answer
2k
views
What is $p$-adic Fourier series?
Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$?
Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion:
$$B_n(\{x\})=-\frac{...
- 12.3k
4
votes
0
answers
144
views
Does a countably generated $\mathbb{Q}$-algebra inject into some $p$-adic field?
Let $K$ be a subfield of $\mathbb{C}$. If $K$ is finitely generated over $\mathbb{Q}$, then $K$ injects into $\mathbb{Q}_p$ for some $p$.
Assume that $K$ is countably generated, i.e., $K= \...
- 71
6
votes
1
answer
410
views
A $p$-adic sum of reciprocals of powers
Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer:
$$
S_{p,k} := \lim_{r\to+\infty} \sum_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k
$$
Convergence is easy to ...
- 32.5k
9
votes
1
answer
896
views
Uniformizer for splitting field of p^{1/p^n} over p-adics
Does anyone know an explicit uniformizer for $\mathbf{Q}_p(\zeta_{p^n}, p^{\frac{1}{p^n}}) / \mathbf{Q}_p$? I was reading the question "adding an n-th root to Q_p" where dke mentions this question but ...
- 1,629
11
votes
1
answer
774
views
2-adic Logarithm and Resistance of n-dimensional Cube
Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=0}^{n-1}\frac1{{n-1\choose k}}.$$
The ...
- 12.3k
1
vote
1
answer
190
views
Hilbert symbols vanishing
Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitve $p^2$-th root of unity $\zeta_{p^2}$. Let $a,b \in E^...
- 11.3k
9
votes
0
answers
408
views
Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$
Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.
Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb ...
- 3,998
4
votes
1
answer
303
views
Transcendence of a ratio of p-adic logarithms
Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$.
If
$$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$
does it follow that ...
- 37k
8
votes
0
answers
245
views
Relation between valuation of p-adic regulator of totally real field and its finite p-unramified abelian extensions
For each prime number $p$ and number field $k$, there exists at least one extension $k_{\infty}/k$ with Galois group isomorphic to $\mathbb{Z}_p$, the cyclotomic $\mathbb{Z}_p$-extension. If $k_p/k$ ...
- 1,521
5
votes
0
answers
262
views
p-adic analogue of self-adjoint operator
Consider the very well-known result that any Hermitian matrix over $\mathbb{C}$, say $T$, admits a decomposition $T = UDU^*$ where $U$ is unitary and $D$ is diagonal with real entries. I am looking ...
8
votes
0
answers
895
views
Lemma in Scholze-Weinstein
In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7:
Lemma: Let $K$ be a ...
- 89
3
votes
1
answer
341
views
Dihedral extension of 2-adic number field
Sorry if the question is too long and maybe elementary.
I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension $K(\...
- 135
6
votes
1
answer
522
views
When do two lattices have the same stabilizer in the diagonal torus?
This is moved from MSE, where I asked and didn't receive an answer (see https://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...
- 1,453
4
votes
1
answer
1k
views
Iwasawa logarithm and analytic continuation
I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.
...
- 539
1
vote
1
answer
475
views
How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$?
Some papers I am reading talk about an "adelic" object $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$ . This has sparked a lot of confusion since I don't know what such a quotient could mean.
A ...
- 22.8k
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
7
votes
1
answer
354
views
$(\varphi, \Gamma)$-modules of finite height
Maybe the answer to my question is obvious.
Let $p$ be a prime $\geq 3$. Let $D$ be an étale $(\varphi, \Gamma)$-module over $A_{\mathbb{Q}_p} = \{ \sum_{n \in \mathbb{Z}} a_n X^n \, \vert \, a_n \in ...
- 477
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
6
votes
0
answers
227
views
Choice of digits for extensions of $\mathbb{Q}_p$
I am interested in writing (in base $p$) elements of the maximal unramified extension $\mathbb{Q}_p^{\mathrm{unr}}$ of $\mathbb{Q}_p$, or (its completion) the field $\mathrm{W}(\mathbb{F}_p^{\mathrm{...
- 32.5k
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
7
votes
2
answers
1k
views
Algebraic $p$-adic integers mod $p$
In studying the papers of Abbes and Saito on ramification theory in the imperfect residue field case, I come to the following questions. Let $\overline{\mathbb{Q}}_p\supset\mathbb{Q}{}^{t}_p\supset \...
- 4,492
1
vote
0
answers
146
views
Class field theory, Ideles class
Let $H$ be a totally complex Galois extension of $\mathbb{Q}$ and $g:G_H \rightarrow \bar{\mathbb{Q}}_p$ be a continuous morphism. By class field thoery we have $\mathrm{Hom}(G_H, \bar{\mathbb{Q}}_p)\...
- 1,066
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
0
votes
0
answers
197
views
Name of some commutative ring akin to $p$-adics
I need help in identifying the naming convention of some commutative ring described below.
Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...
- 103