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
103 questions with no upvoted or accepted answers
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Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 631
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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
11
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807
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Torelli-like theorem for K3 surfaces on terms of its étale cohomology
Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow \...
10
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248
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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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834
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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 ...
- 989
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408
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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
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Differences in tree picture of ${\bf Q}_p$, $\overline{{\bf Q}_p}$, ${\bf C}_p$, $\Omega_p$
I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value ...
- 441
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389
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Can the p-adic be countable?
Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
- 695
8
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584
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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}...
- 1,479
8
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245
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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
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895
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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 ...
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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 ...
- 171
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489
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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 ...
- 1,338
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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
$\mathcal{O}_K^...
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7
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The character of a separable degree-$p$ extension of a local field of residual characteristic $p$ ?
Let $p$ be a prime number and $F$ a finite extension of ${\mathbf Q}_p$ or of ${\mathbf F}_p((t))$. I'm going to define a natural map from the set ${\mathcal S}_p(F)$ of degree-$p$ separable ...
- 18.7k
6
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228
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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 ...
- 1,949
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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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698
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Jacobson radical of a tensor product
Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
- 6,349
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Specifying cokernels of all powers of $p$-adic matrix
Given a matrix $A \in M_d(\mathbb{Z}_p)$ (with nonzero determinant), viewed as a map $\mathbb{Z}_p^d \to \mathbb{Z}_p^d$, I am interested in the sequence of abelian $p$-groups $\{coker(A^n)\}_{n \geq ...
- 568
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227
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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
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514
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Finite extensions of $\mathbb Q_p$
Is there any finite extension of $\mathbb Q_p$ which is not the completion of a finite extension of $\mathbb Q$ at some place over $p$?
Analogously in equicharacteristic, if $k=\overline {\mathbb F_p}$...
- 277
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2k
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Newton Method in $p$-adic case
The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentiable assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic integers,...
- 1,109
5
votes
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197
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Bezout-type theorem for $p$-adic analytic plane curves
Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
5
votes
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181
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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 \...
- 159
5
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328
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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
5
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262
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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 ...
5
votes
0
answers
758
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maximal abelian extension of quadratic extension of $\mathbb Q_p$
I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,...
- 51
4
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90
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Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
- 6,622
4
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66
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields
Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
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170
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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 $\...
- 930
4
votes
0
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102
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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 ...
- 335
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0
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872
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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 $...
- 6,180
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306
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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
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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
4
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0
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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
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Structure of modules over Iwasawa algebra $\mathbb{Z}_p[[T]]$ when taken mod $p$
Let $A \in M_n(\mathbb{Z}_p)$ be a nonsingular matrix which is nilpotent mod $p$, so $A^r \in pM_n(\mathbb{Z}_p)$ for some $r$. Then $\mathbb{Z}_p[[T]]$ acts on $\mathbb{Z}_p^n$ with $T$ acting by $A$....
- 568
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0
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144
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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
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174
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Simultaneously using the real and 2adic norms
In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...
user13113
4
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153
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Finiteness of the set of $\mathbb{Q}_p$-rational periodic points
The statement I am concerned with is this:
Let $\varphi : \mathbb{P}^r_{\mathbb{Z}_p} \to \mathbb{P}^r_{\mathbb{Z}_p}$ be a morphism of degree higher than one. Then the set of $\mathbb{Q}_p$-...
- 13.8k
3
votes
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Problem Deducing the value of Quadratic Hilbert Symbol from Explicit Formulas
This question concerns the explicit law for the Hilbert Symbol given in Sur les lois de réciprocfites explicites I by Henniart. I am trying to deduce the classical value of the Hilbert Symbol in $\...
- 171
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votes
0
answers
60
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What circumstances guarantee a p-adic affine conjugacy map will be a rational function?
Let $\Bbb Q_p$ be a p-adic field and let any element $x$ of $\Bbb Q_p$ be associated with a unique element of $\Bbb Z_p$ via the quotient / equivalence relation $\forall n\in\Bbb Z:p^nx\sim x$
Then in ...
- 308
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votes
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answers
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Algebraic properties of Witt vectors $W(K^{\flat\circ})$, $K$ a characteristic 0 perfectoid field
Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \...
- 658
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votes
0
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145
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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 ...
- 7,746
3
votes
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answers
281
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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 $\mathcal{O}_{\mathcal{C}}$. ...
- 519
3
votes
0
answers
128
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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}^{\...
- 2,197
3
votes
0
answers
191
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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 $...
- 313
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votes
0
answers
265
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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 ...
- 7,182
3
votes
0
answers
163
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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
$$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$
...
- 17.9k
3
votes
0
answers
94
views
Projective limit of copies of same group w.r.t. some fixed endomorphism
In our study of automorphism groups of transcendental field extensions, we have encountered the situation where we have a group $F$ together with an endomorphism $\alpha \colon F \to F$, resulting in ...
- 6,614
3
votes
0
answers
118
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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^...
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