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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What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2 \mathbb{Z}_p$?
Let $p \geq 2$ be prime and $K=\mathbb{Q} (\zeta_p),~ \zeta^{p-1}=1$ with ring of integers $\mathcal{O}_K$. We denote $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ ...
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$p$-adic L function of an odd Dirichlet character
Apologies for a naive question (especially for Iwasawa theorists): it is well-known
and trivial to prove that the usual (elementary) construction of $p$-adic L functions
attached to odd Dirichlet ...
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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 $r = 0$
$N(r+s) \leq N(r) + N(...
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$\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$?
$\newcommand\tetra[2]{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2), it has been pointed out how my ...
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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:...
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Does $p$-adic Baker theorem holds in the given case?
Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, and $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$. Let $U_p$ be the units $(1+\mathfrak{m})$ of $\...
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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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A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture
A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
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$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 ...
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$p$-adic $L$-functions and congruence of $L$-values
I am reading about $p$-adic $L$-functions and I have one question in mind.
To start with, I will write a proof I've learned of a congruence of $L$-values:
Theorem: Let $p\geq5$ be a prime, $\alpha\...
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Nygaard filtration on Fontaine's period ring
Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. ...
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Existence of intermediate field extensions for tamely ramified p-adic extensions
Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
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Sums of powers of measures of $p$-adic balls
Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...
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Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...
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A confusion about power series and p-adic measures
In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:
Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{...
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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)$ ...
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On $p$-adic arithmetic softwares
I would like to automate a huge amount of computation that involves basic arithmetic operations with $p$-adic numbers. I have found a Mathematica package for it, but it is old and acts quite ...
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Galois cohomology with coefficients in the integers of the Lubin-Tate extension
Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
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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 ...
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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$. ...
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Non-existence of "higher" Artin map
So rank $1$ local Langlands is special in as that it is given by the Artin map
$$\text{GL}_1(K)\to G_K^{ab},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\...
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How extension $\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ looks like?
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 know what
$\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ is .
At first I tried to prove ...
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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 $p$ ...
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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 ...
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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 ...
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defining the upper ramification numbering
Given a local field $K$ with absolute Galois group $\Gamma$. Is it "possible" to define the upper numbering on $\Gamma$ without using the lower numbering?
In other words, given $\gamma \in \...
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Faster computation of p-adic log
As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting).
When it comes to computing $\log P(x)$, one may use the formula
$$
(\log P)' = \...
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Describing the Gamma-transform explicitly in terms of power series
The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma_{\alpha}(s) = \int_{\mathbf{...
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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 ...
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Some variants of Artin's primitive root conjecture
Artin's primitive root conjecture asserts that if an integer $a \ne -1$ is not a perfect square then $a$ is a primitive root mod $p$ for infinitely many primes $p$. This conjecture is still open.
An ...
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Are the $p$-adic digits of roots of unity equidistributed?
I was looking at the $p$-adic expansions of roots of unity in $\mathbf{Z}_p$, and I noticed that the digits tended to be equidistributed among the numbers $\{0, 1, \dots, p-1 \}$. I wanted to ask if ...
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Elementary aspects of The Fargues-Fontaine curve
To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius ...
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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{\...
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Sign error in $\pm$-parts of modular symbols?
I am trying to connect the definition of $\pm$-modular symbols given in [Pollack, pg. 529] and [MTT,pg. 11] to those appearing in [Greenberg-Stevens, pg. 200 in #20 here], but I can't seem to ...
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Decomposition of primes in cyclotomic extensions and their ramifications
Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$.
So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...
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p-adic analogue of octonions
There are the complex p-adic numbers.
But what is the p-adic analogue of the Cayley–Dickson construction?
Or more important: What is the p-adic analogue of the octonions?
It would be nice if the (unit)...
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Algebraic numbers in all $\mathbb Q_p$ [duplicate]
Do there exist non-rational algebraic numbers that belong to $\mathbb Q_p$ for all prime $p$? If yes, can one characterize them?
I spent several days for the first question, and I found nothing. The ...
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Is there a classification of the $p$-adic normed division algebras?
A normed division algebra over $\mathbb{R}$ is a pair $(A,\lVert{-}\rVert)$ with
$A$ an $\mathbb{R}$-algebra with a unit $1_A$;
$\lVert{-}\rVert\colon A\to\mathbb{R}_{\geq0}$ a norm on $A$;
such ...
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To justify the intuition about #$E(\Bbb Q_p)$＝$∞$
Let $E$ be an elliptic curve on $\Bbb Q_p$.
$E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points.
How to prove #$E(\Bbb Q_p)$＝$∞$ directly ?
According to Silverman's book 'the ...
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Method to solve modular quadratic polynomial [duplicate]
If $q$ is a prime what is the best method to compute roots of a quadratic polynomial $f(x)\equiv0\bmod q^2$ which is of form $x^2+bx+c\equiv0\bmod q^2$ where $b^2-4c\equiv0\bmod q$ and $gcd(b,q)=1$ ...
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Is the p-adic density of the image of a polynomial always rational?
This question was previously posted here on MSE.
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. For $n\in\mathbb N$, let $I_n$ be the number of integers $i\in\{1,\...
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Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?
When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
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Scholze and Weinstein's $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}_p$
In their Berkeley Lectures, to motivate the introduction of Diamonds, Scholze and Weinstein discuss what should be the definition of $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}...
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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
$$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$
...
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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 ...
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Why are the $p$-adic $L$-functions for a modular form with $a_p=0$ conjugates?
I have a question about the proof of Theorem 3.5 in Pollack's 2003 paper On the $p$-adic L-function of a Modular Form at a Supersingular Prime.
The setup is as follows. Fix an eigenform $f\in S_k(N,\...
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Characters of p-adic units
Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...
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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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A family of Diophantine equations with no integer solutions but solutions modulo every integer
Selmer's curve is the equation $3x^3 +4y^3 +5z^3=0$. This equation is famous for having non-trivial solutions in every completion of $\mathbb{Q}$ but only having the trivial solution in the rationals. ...
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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 ...
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