p-adic completion

[[!redirects p-completion]]

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## Idea

$p$-completion is to [[p-adic homotopy theory]] as [[rationalization]] is to [[rational homotopy theory]].

For more see at _[[formal completion]]_.

### Of abelian groups, rings and modules

For $A$ an [[abelian group]] (or [[commutative ring]]) and $p$ a [[prime number]], the _$p$-completion_ of $A$ is the [[limit]]

$$
  A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_{n \geq 1} A/(p^n A)
  \,.
$$

(e.g. [May Ponto, 10.1.1](#MayPonto)) For more see at _[[formal completion]]_.

$A$ is called _$p$-complete_ if the canonical [[homomorphism]] $A \to A_p^\wedge$ is an [[isomorphism]].

### Of a homotopy type

(...) ([e.g. May-Ponto, 10.2](#MayPonto))

## Properties

The _[[fracture theorem]]_ says that under mild conditions a (stable) homotopy type decomposes into its [[rationalization]] and its $p$-completions.

## Examples

For $A = \mathbb{Z}$ the [[integers]], the $p$-completion is the [[p-adic integers]]. (Notice that here traditionally one writes $\mathbb{Z}_p = \mathbb{Z}_p^\wedge$.)

More generally, if $A$ is [[finitely generated object|finitely generated]], then $A_p^\wedge \simeq A\otimes \mathbb{Z}_p$. (e.g. [May Ponto, p. 154](#MayPonto))

## Related concepts

* [[p-adic integers]]

* [[completion of a group]]

* [[completion of a ring]], [[analytic completion of a ring]]

* [[p-localization]]

* [[p-adic homotopy theory]]

* [[rationalization]], [[rational homotopy theory]]

* [[fracture theorem]]

* [[nilpotent homotopy type]], [[finite homotopy type]]

* [[nilpotent completion of spectra]]

* [[L-complete module]]

## References

Classical accounts:

* {#Sullivan05} [[Dennis Sullivan]], _Geometric topology: localization, periodicity and Galois symmetry_, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface
by [[Andrew Ranicki]] ([pdf](http://www.maths.ed.ac.uk/~aar/books/gtop.pdf))

* {#BousfieldKan71} [[Aldridge Bousfield]], [[Daniel Kan]], *Localization and completion in homotopy theory*, Bull. Amer. Math. Soc. **77** 6 (1971) 1006-1010 [[doi:10.1090/S0002-9904-1971-12837-9](https://doi.org/10.1090/S0002-9904-1971-12837-9), [pdf](https://www.ams.org/journals/bull/1971-77-06/S0002-9904-1971-12837-9/S0002-9904-1971-12837-9.pdf)]

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, **304** Springer (1972) [[doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4)]

* {#MayPonto} [[Peter May]], [[Kate Ponto]], chapters 7 and 8 of _More concise algebraic topology: Localization, completion, and model categories_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf))

See also

* [[Doug Ravenel]], chapter 2, def. 2.1.14 _[[Complex cobordism and stable homotopy groups of spheres]]_

* {#Rognes12} [[John Rognes]], section 4.6 of _The Adams spectral sequence_, 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

[[!redirects p-completions]]

[[!redirects p-complete ring]]
[[!redirects p-complete rings]]

[[!redirects p-complete homotopy type]]
[[!redirects p-complete homotopy types]]


p-adic complex number

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## Idea

For $p$ a [[prime number]], the [[field]] of _complex $p$-adic numbers_ $\mathbb{C}_p$ is to the [[p-adic numbers]] $\mathbb{Q}_p$ as the [[complex numbers]] $\mathbb{C}$  are to the [[real numbers]].

## Definition

First observe that the ordinary [[complex numbers]] $\mathbb{C}$ may be characterized as follows:

the standard [[absolute value]] ([[norm]]) on the [[rational numbers]] $\mathbb{Q}$ uniquely extends to an [[algebraic closure]] $\bar \mathbb{Q}$, and the [[complete field|completion]] is the complex numbers.

In direct analogy with this:

for $p$ a [[prime number]] and $\mathbb{Q}_p$ the corresponding [[non-archimedean field]] of [[p-adic rational numbers]], then the [[complete field|completion]] of any [[algebraic closure]] $\bar \mathbb{Q}_p$ is the field of complex $p$-adic numbers $\mathbb{C}_p$.

Notice that the completion of the algebraic closure of a [[normed field]] is still algebraically closed ([Bosch-Guntzer-Remmert 84, prop. 3.4.1.3](#BoschGuntzerRemmert84)). See also at _[normed field -- relation to algebraic closure](http://ncatlab.org/nlab/show/normed+field#RelationToAlgebraicClosure)_.

## References

* L. C. Washington, _Introduction to Cyclotomic Fields_, Springer-Verlag, New York.

* PlanetMath, _[complex p-adic numbers](http://planetmath.org/complexpadicnumbers)_

* {#BoschGuntzerRemmert84} S. Bosch, U. Guntzer, and R. Remmert, _Non-Archimedean analysis_, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984.

[[!redirects p-adic complex numbers]]

[[!redirects complex p-adic number]]
[[!redirects complex p-adic numbers]]


p-adic differential equation

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## Idea

A **p-adic differential equation** is a [[differential equation]] in which the role of the [[real numbers]] is played by the [[field]] of [[p-adic numbers]] for some prime $p$.

According to [Kedlaya 10](#Kedlaya10), the study of [[zeta functions]] via $p$-adic analysis lends itself much more readily to numerical computation than do methods employing [[étale cohomology]]:

>...it is not straightforward to describe étale cohomology to a computer well enough that the computer can make calculations. ([Kedlaya 10, p.5](#Kedlaya10))

## Related entries

* [[arithmetic cryptography]]
* [[rigid cohomology]]

## References

* {#Kedlaya10} [[Kiran Kedlaya]], _p-adic Differential Equations_, CUP, 2010, ([doi](https://doi.org/10.1017/CBO9780511750922))

[[!redirects p-adic differential equations]]


p-adic geometry

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## Idea

The special case of [[non-archimedean analytic geometry]] based on [[p-adic numbers]].

## Examples

* [[adic space]]

* [[non-archimedean analytic geometry]]

* [[Berkovich analytic space]]

## Related concepts

* [[p-adic homotopy]]

* [[p-adic cohomology]]

* [[p-adic physics]]

## References

Survey is in

* {#LeStum12} [[Bernard Le Stum]], _One century of $p$-adic geometry -- From Hensel to Berkovich and beyond_, talk notes, June 2012 ([pdf](http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf))

* {#Scholze17} [[Peter Scholze]], _$p$-adic geometry_, ICM report ([arXiv:1712.03708](https://arxiv.org/abs/1712.03708))

* {#ScholzeLectures14} [[Peter Scholze]], [[Jared Weinstein]], _Berkeley lectures on $p$-adic geometry_, [pdf](http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf).


p-adic Hodge theory

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## Idea

A surprising parallel between classical [[Hodge theory]] and the theory of [[Galois representations]] was noted in the beginning of the 60's; shortly [[John Tate|Tate]] and [[Alexander Grothendieck|Grothendieck]] attempted to understand the topology of p-adic [[algebraic variety|varieties]] in light of this analogy.
A series of precise conjectures about p-adic Hodge theory was formulated by [[Jean-Marc Fontaine|Fontaine]] at the beginning of the 80's; three different proofs were proposed by [[Gerd Faltings|Faltings]], [[Wieslawa Niziol|Niziol]] and [[Takeshi Tsuji|Tsuji]].
A significantly simpler approach was developed recently in works of [[Alexander Beilinson|Beilinson]] and [[Bhargav Bhatt|Bhatt]].

_This paragraph is translated from the abstract of ([Beilinson's Yaroslavly' lectures](#BeilinsonLectures2014))._

## Overview

_p-adic Hodge theory_ is the study of properties of p-adic (étale, de Rham, logarithmic cristalline) [[cohomology]] (and [[motives]]) of non-archimedean [[analytic spaces]]. The $p$-adic Hodge structure of a (proper or semi-stably compactified) p-adic analytic variety is essentially given by a relation between three important invariants of the given variety:

* p-adic de Rham cohomology equipped with the Hodge filtration,

* log-cristalline cohomology (of a potentially semi-stable model) equipped with its Frobenius and monodromy map,

* [[pro-étale cohomology]] of the extension of the given variety to a `chosen` algebraic closure of the base field, together with its natural action of the Galois group.

There is an "evident" isomorphism between p-adic de Rham cohomology and log-cristalline cohomology (both are given by a kind of differential calculus).

The main theorem of p-adic Hodge theory is that the datum of these two cohomologies and their relation
on the one side, and of pro-étale cohomology on the other side, mutually determine each other (once
equipped with their natural additional structures).

This result has been conjectured first by Grothendieck in the case of p-divisible
groups. He called it the `mysterious functor`. A conjecture generalizing this
statement to other kinds of ---algebraic--- p-adic varieties was formulated by
Jannsen in a particular case, and then grounded on the setting of [[p-adic periods]]
by Fontaine. After these important Breakthrough, the theory was fully developed by a long list of contributors (see `periodes p-adiques`, Astérisque for a first list). The comparison theorem was proved by many authors in various particular cases, and then in full generality (for algebraic varieties) by Faltings
using [[almost mathematics]].

During the last few years, there were interesting new developments.

If one works with rational coefficients (no torsion), one may get the isomorphism between both
structures in a geometric way (in an algebraic setting),
using Beilinson's approach, through (derived de Rham cohomology)[[de Rham complex]].

It is also possible to use Scholze's [[perfectoid spaces]]
and Faltings' [[almost mathematics]] to give a very elegant
proof of this theorem for proper (or more generally, semi-stably compactified) p-adic
(strict, i.e., rigid) analytic varieties. The fact that the classical result extends
to the analytic setting is essentially due (up to difficulties related to the resolution in finite characteristic, that are dealt with using Faltings' methods) to the fact that (proper) rigid analytic varieties always
have an integral model, given by a formal scheme over the ring of integers of the given
non-archimedean field.
The same proof may also extend to the non-strict situation by using a convenient base
extension, but this work has not yet been done.

The very important case of torsion coefficients has also been
addressed by Scholze in his setting, and by Bhatt, who gave a useful refinement
of Beilinson's construction. Remark that almost mathematical tools seem to be quite well adapted to the study of completed integral cohomology.

Scholze's original method involves the use of Witt vectors (already used in
Fontaine's definition of period rings), that were also
studied by Kedlaya and Liu in this Hodge theoretic context. This Witt vector
approach are interesting, but they seem to have the drawback of being
quite hard to globalize (including the archimedean norm in a context of
[[global Hodge theory]]).

It is quite clear to the experts that a nice way to overcome the difficulties that appear
when one uses Witt vectors would be to combine directly the (analytic) `derived de Rham`
approach of Beilinson and Bhatt to the `perfectoid and almost mathematics` approach of
Faltings and Scholze. Both approaches may be extended to the analytic setting using
overconvergent derived analytic spaces (see [[global analytic geometry]]).

One may say that almost geometrical derived methods could be useful to study integral completed cohomology, while usual derived geometric methods are quite
well adapted to the study of torsion phenomena in finite characteristic.

Indeed, as explained by Bhatt at the end of his paper, one may use derived de Rham cohomology
over $\Z$ to get a (non-archimedean) period ring isomorphism for an [[arithmetic variety]].

## Hodge-Tate decomposition

An early result in $p$-adic Hodge theory is the Hodge-Tate decomposition, which is a $p$-adic analogue of the Hodge decomposition in classical [[Hodge theory]]. Let $X$ be a smooth projective variety over $\mathbb{Q}_{p}$ (or more generally some finite extension of it). Let $\mathbb{C}_{p}$ be the [[p-adic complex numbers]]. Then we have

$$H^{k}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{Q}}^{j})\otimes_{\mathbb{Q}}\mathbb{C}_{p}(-j)$$

## Comparison theorems

Central to modern $p$-adic Hodge theory are the comparison theorems that relate the [[de Rham cohomology]] and the [[étale cohomology]] of a smooth projective variety $X$ over $\mathbb{Q}_{p}$ (or more generally some finite extension of it), using the machinery of period rings, whose construction we will discuss in the next section.

For instance, tensoring with the de Rham period ring $B_{dR}$ gives us the following isomorphism between the de Rham and $p$-adic etale cohomology of $X$:

$$H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}=H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}$$

The ring $B_{dR}$ is equipped with a filtration and a Galois action. We have a way of recovering the de Rham cohomology from the $p$-adic etale cohomology by taking Galois invariants:

$$H_{\mathrm{dR}}^{i}(X)=(H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$$

Recovering the $p$-adic etale cohomology from the de Rham cohomology involves a different period ring, the crystalline period ring $B_{cris}$ (which is equipped with a Frobenius $\varphi$ in addition to a filtration and Galois action), and is only possible if $X$ has an integral model:

$$H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})= \mathrm{Fil}^{0}(H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{cris}})^{\varphi=1}$$

## Construction of period rings

We now show how to construct $B_{dR}. $Let $\mathcal{O}_{\mathbb{C}_{p}}$ be the ring of integers of $\mathbb{C}_{p}$. We take the tilt $\mathbb{O}_{\mathbb{C}_{p}}^{\flat}$ of $\mathbb{O}_{\mathbb{C}_{p}}$, and take Witt vectors. The resulting ring $W(\mathbb{O}_{\mathbb{C}_{p}}^{\flat})$ is also called $A_{inf}(\mathcal{O}_{\mathbb{C}_{p}})$. It comes with a canonical map $\theta:A_{inf}(\mathcal{O}_{\mathbb{C}_{p}})\to \mathcal{O}_{\mathbb{C}_{p}}$. Inverting $p$ and taking the completion with respect to $\theta$ gives us the ring $B_{dR}^{+}$. There is a special element $t\in B_{dR}^{+}$ which we think of as the logarithm of the element $(1,\zeta,\zeta^{1/p},\ldots)$. Then we define $B_{dR}=B_{dR}^{+}[1/t]$.

Next we show how to construct $B_{cris}$. Once again we take $A_{inf}(\mathcal{O}_{\mathbb{C}_{p}})$ and invert $p$. Instead of completing with respect to $\theta$, as in the construction of $B_{dR}$, we take a generator $\omega$ of its kernel, and consider the ring $B_{cris}^{+}$ whose elements are power series of the form $\sum a_{n}\omega^{n}$ where the $a_{n}$'s are elements of $A_{inf}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ which converge to $0$ as $n$ approaches infinity, with respect to the topology of $A_{inf}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$. Once again there will be an element $t$ as in the construction of $B_{dR}$; we define $B_{cris}=B_{cris}^{+}[1/t]$.

## Classification of $p$-adic Galois representations

We can abstract the concepts discussed above and take a p-adic Galois representation $V$ (without knowing, say, if it comes from some cohomology theory). We can then define

$$V_{dR}=(V\otimes B_{dR})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$$

If the dimension of $V$ is equal to the dimension of $V_{dR}$, we say that $V$ is _de Rham_. We can also apply this construction with $B_{cris}$ instead of $B_{dR}$; if the dimension stays the same we say that $V$ is _crystalline_.

## $p$-adic Hodge theory for rigid analytic spaces

[[Peter Scholze]] has developed a version of $p$-adic Hodge theory for [[rigid analytic spaces]] in [Scholze13](#Scholze13). It relies on the construction of period sheaves and develops a $p$-adic version of the [[Riemann-Hilbert correspondence]] as an intermediate step.

## Related concepts

* [[Galois representation]]

* [[Fargues-Fontaine curve]]

* [[prismatic cohomology]]

## References

### Overview

[[Jacob Lurie]], _Lecture 1: Overview_, lecture notes from a learning seminar on the [[Fargues-Fontaine curve]], [pdf](https://www.math.ias.edu/~lurie/ffcurve/Lecture1-Overview.pdf)

### Introductory references

[[Laurent Berger]], _An Introduction to the Theory of $p$-adic Representations, [pdf](http://perso.ens-lyon.fr/laurent.berger/articles/article05.pdf)

[[Oliver Brinon]] and [[Brian Conrad]], _CMI Summer School Notes on $p$-adic Hodge Theory, [pdf](https://math.stanford.edu/~conrad/papers/notes.pdf)

Niziol's [[algebraic K-theory|K-theoretic]] proof:

* [[Wiesława Nizioł]], _Crystalline conjecture via $K$-theory_, Annales scientifiques de l'École Normale Supérieure 31.5 (1998): 659-681. [web](http://eudml.org/doc/82474).

### Beilinson-Bhatt approach

* [[Alexander Beilinson]], _p-adic periods and derived de Rham cohomology_, [arXiv:1102.1294](http://arxiv.org/abs/1102.1294).

* [[Alexander Beilinson]], _On the crystalline period map_, [arXiv:1111.3316](http://arxiv.org/abs/1111.3316).

* [[Bhargav Bhatt]], _p-adic derived de Rham cohomology_, [arXiv:1204.6560](http://arxiv.org/abs/1204.6560).

* [[Luc Illusie]], _Around the Poincaré lemma, after Beilinson_ (Preliminary notes), 2013, [pdf](http://www.math.u-psud.fr/~illusie/derived-deRham3.pdf).

* {#BeilinsonLectures2014} [[Alexander Beilinson]], _p-adic Hodge theory_, lectures from Yaroslavl' summer school 2014, [videos](http://bogomolov-lab.ru/SHKOLA2014/talks/beilinson.html).

### $p$-adic Hodge theory for rigid analytic spaces

* {#Scholze13} [[Peter Scholze]], _p-adic Hodge theory for rigid-analytic varieties_, Forum of Mathematics, Pi, 1, e1 2013. [arXiv:1205.3463](https://arxiv.org/abs/1205.3463)

### $p$-adic absolute Hodge cohomology

* [[Frédéric Déglise]], [[Wiesława Nizioł]], _On $p$-adic absolute Hodge cohomology and syntomic coefficients, I_, [arXiv:1508.02567](http://arxiv.org/abs/1508.02567).


p-adic homotopy theory

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## Idea

In _$p$-adic homotopy theory_ one studies, for any [[prime number]] $p$, [[simply connected]] [[homotopy types]] (of [[topological spaces]], hence [[∞-groupoids]]) all of whose [[homotopy groups]] have the structure of  (finitely generated) [[modules]] over the [[p-adic integers]] $\mathbb{Z}_p$ -- the _$p$-adic homotopy types_. The central theorem ([Mandell 01](#Mandell01)) says that the [[(∞,1)-category]] on $p$-adic homotopy types is [[sub-(∞,1)-category|faithfully embedded]] into the [[(∞,1)-category]] of [[E-∞ algebras]] over an [[algebraically closed field]] of [[characteristic]] $p$ ([[Rational and p-adic Homotopy Theory|Lurie, p. 70, cor. 3.5.15]]).

In this way $p$-adic homotopy theory is directly analogous to _[[rational homotopy theory]]_ with the [[rational numbers]] $\mathbb{Q}$ replaced by the [[p-adic integers]].

The _[[fracture theorem]]_ says that under mild conditions, homotopy theory may be decomposed in a precise sense into rational homotopy theory and $p$-adic homotopy theory for each prime $p$.

## Related concepts

* [[p-adic numbers]]

* [[p-completion]], [[p-localization]]

* [[mod p Whitehead theorem]]

* [[p-adic geometry]]

* [[p-adic cohomology]]

* [[p-adic physics]]

* [[rational homotopy theory]], [[real homotopy theory]]

## References

* {#Mandell01} [[Michael Mandell]], _$E_\infty$-algebras and $p$-Adic homotopy theory_, Topology 40 (2001), no. 1, 43-94. ([pdf](http://hopf.math.purdue.edu/Mandell/einf.pdf))

  surveyed in: _Algebraic models in $p$-adic homotopy theory_, YTM13, 2013 ([pdf](https://mmandell.pages.iu.edu/YTM/YTC3-handout.pdf))

* [[Jacob Lurie]], _$p$-Adic homotopy theory_ ([pdf](http://www-math.mit.edu/~lurie/917notes/Lecture27.pdf))

* {#Lurie} [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

[[!redirects p-adic homotopy theories]]

[[!redirects p-adic homotopy]]


p-adic integer

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## Idea
 {#Idea}

For each [[prime number]] $p$ the [[ring]] of _$p$-adic integers_ $\mathbb{Z}_p$ is the [[formal completion]] of the ring $\mathbb{Z}$ at the [[prime ideal]] $(p)$. Geometrically this means that $\mathbb{Z}_p$ is the [[ring of functions]] on a [[formal neighbourhood]] of $p$ inside [[Spec(Z)]] (this is discussed in more detail [below](#AsFormalNeighbourhoodOfPrime)). Algebraically it means that the elements in $\mathbb{Z}_p$ look like [[formal power series]] where the formal variable is the prime number $p$.

## Definition

For any [[prime number]] $p$, the [[ring]] of $p$-**adic integers** $\mathbb{Z}_p$ (which, to avoid possible confusion with the ring $\mathbb{Z}/(p)$ used in [[modular arithmetic]], is also written as $\widehat{\mathbb{Z}}_p$) may be described in one of several ways:

1. To the person on the street, it may be described as (the ring of) numbers written in base $p$, but allowing infinite expansions to the left. Thus, numbers of the form
$$\sum_{n \geq 0} a_n p^n$$
where $0 \leq a_n \lt p$, added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.

2. More abstractly, it is the [[limit]] $\underset{\leftarrow}{\lim} \mathbb{Z}/(p^n)$, in the [[category]] of (unital) [[ring|rings]], of the [[diagram]]
$$ \ldots \to \mathbb{Z}/(p^{n+1}) \to \mathbb{Z}/(p^n) \to \ldots \to \mathbb{Z}/(p) .$$
This is also a limit in the category of [[topological rings]], taking the rings in the diagram to have [[discrete topologies]].

3. Alternatively, it is the [[metric completion]] of the ring of [[integers]] $\mathbb{Z}$ with respect to the $p$-adic [[absolute value]]. Since addition and multiplication of integers are [[uniform space|uniformly continuous]] with respect to the $p$-adic [[absolute value]], they extend uniquely to a uniformly continuous addition and multiplication on $\mathbb{Z}_p$. Thus $\mathbb{Z}_p$ is a [[topological ring]].

4. Also $\mathbb{Z}[ [ x ] ]/(x-p)\mathbb{Z}[ [ x ] ]$, see at _[[analytic completion]]_.

Hence one also speaks of the _$p$-[[adic completion]]_ of the integers.  See [[completion of a ring]] (which generalizes 2&3).

There is also this characterization:

+-- {: .num_lemma #pAdicIntegersAspExtensionofFpByThemselves}
###### Lemma

There is a [[short exact sequence]]

$$
  0
    \to
  \mathbb{Z}_{p}
    \overset{p \cdot (-)}{\longrightarrow}
  \mathbb{Z}_{p}
    \longrightarrow
  \mathbb{Z}/p\mathbb{Z}
    \to
  0
  \,.
$$

=--

+-- {: .proof}
###### Proof

Consider the following [[commuting diagram]]

$$
  \array{
     \vdots && \vdots && \vdots
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p^3\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^4 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p^2\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^3 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^2 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     0
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
  }
  \,.
$$

Each horizontal sequence is exact. Taking the [[limit]] over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.

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## Properties

### Topology

The ring of $p$-adic integers has the following properties:

* As a [[topological space]], it is [[compact space|compact]], [[Hausdorff space|Hausdorff]], and [[totally disconnected space|totally disconnected]] (i.e., is a [[Stone space]]). Moreover, every point is an [[accumulation point]], and there is a countable basis of [[clopen set|clopen sets]] -- a Stone space with these properties must be [[homeomorphism|homeomorphic]] to [[Cantor space]].

* As a [[topological group]] under addition, it is therefore an [[almost connected group]]. As an [[abelian group|abelian]]  [[compact group]], it is [[Pontryagin duality|Pontryagin dual]] to the [[Prüfer group|Prüfer]] $p$-group as [[discrete group]].

### Relation to profinite completion of the integers

+-- {: .num_example #ProfCompletionOfIntegers}
###### Example

The [[profinite completion of the integers]] is

$$
  \widehat {\mathbb{Z}}
    \coloneqq
  \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z})
  \,.
$$

This is [[isomorphism|isomorphic]] to the [[product]] of the $p$-adic integers for all $p$

$$
  \widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p
  \,.
$$

=--

([e.g. Lenstra, example 2.2](#Lenstra))

+-- {: .num_defn }
###### Definition

The ring of integral [[adeles]] $\mathbb{A}_{\mathbb{Z}}$ is the [[product]]
of the profinite completion $\widehat{\mathbb{Z}}$ of the integers,
example \ref{ProfCompletionOfIntegers},
with the [[real numbers]]

$$
  \mathbb{A}_{\mathbb{Z}}
  \coloneqq
  \mathbb{R} \times \widehat{\mathbb{Z}}
  \,.
$$

The [[group of units]] of the ring of adeles is called the group of [[ideles]].

=--

### Pontrajgin duality to Prüfer $p$-group

Under [[Pontryagin duality]], the [[abelian group]] underlying $\mathbb{Z}_p$ maps to the [[Prüfer p-group]] $\mathbb{Z}[p^{-1}]/\mathbb{Z}$, see at _[[Pontryagin duality for torsion abelian groups]]_.

$$
  \array{
     &\mathbb{Z}[p^{-1}]/\mathbb{Z}
     &\hookrightarrow&
     \mathbb{Q}/\mathbb{Z}
     &\hookrightarrow&
     \mathbb{R}/\mathbb{Z}
     \\
     {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow
     \\
     &\mathbb{Z}_p
     &\leftarrow&
     \hat \mathbb{Z}
     &\leftarrow&
     \mathbb{Z}
  }
$$

### As the formal neighbourhood of a prime
 {#AsFormalNeighbourhoodOfPrime}

The [[formal spectrum]] $Spf(\mathbb{Z}_p)$ of $\mathbb{Z}_p$ may be understood as the [[formal neighbourhood]] of the point corresponding to the [[prime]] $p$ in the [[prime spectrum]] $Spec(\mathbb{Z})$ of the [[integers]]. The inclusion

$$
  \{p\}
  \hookrightarrow
  Spf(\mathbb{Z}_p)
  \hookrightarrow
  Spec(\mathbb{Z})
$$

is the [[Isbell duality|formal dual]] of the canonical [[projection]] maps $\mathbb{Z}\to \mathbb{Z}_p\to \mathbb{Z}/(p)$.

This plays a central role for instance in the [[function field analogy]].
It is highlighted for instance in ([Hartl 06, 1.1](Hartl06), [Buium 13, section 1.1.3](#Buium13)). See also at _[[arithmetic jet space]]_ and at _[[ring of Witt vectors]]_.

[[!include infinitesimal and local - table]]

## Related notions

* $p$-[[p-adic number|adic number]], [[adele]].

* [[p-adic complex number]]

* [[Z-infinity-module]]

* [[p-completion]]

* [[local-global principle]]

* [[p-adic homotopy]]

* [[ℓ-adic cohomology]]

* [[p-adic geometry]]

* [[p-adic physics]]

* [[motivic integration]]

## References

Introductions and surveys include

* {#Sullivan70} [[Dennis Sullivan]], pp. 9 of _Localization, Periodicity and Galois Symmetry_ (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer ([pdf](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf))

* Bernard Le Stum, _One century of $p$-adic geometry -- From Hensel to Berkovich and beyond_ talk notes, June 2012 ([pdf](http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf))

* {#Lenstra} [[Hendrik Lenstra]], _Profinite groups_ ([pdf](http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf))

The [[synthetic differential geometry]]-aspect of the $p$-adic numbers is highlighted for instance in

* {#Hartl06} [[Urs Hartl]], _A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic_ ([arXiv:math/0607182](http://arxiv.org/abs/math/0607182))

* {#Buium13} [[Alexandru Buium]], _Differential calculus with integers_ ([arXiv:1308.5194](http://arxiv.org/abs/1308.5194))

[[!redirects p-adic integer]]
[[!redirects p-adic integers]]

[[!redirects adic integer]]
[[!redirects adic integers]]

[[!redirects ring of p-adic integers]]
[[!redirects rings of p-adic integers]]


p-adic integration

+-- {: .rightHandSide}
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###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

## Related concepts

* [[motivic integration]]

## References

* [[Thomas Hales]], _Can p-adic integrals be computed?_ ([arXiv:math/0205207](http://arxiv.org/abs/math/0205207))


p-adic local Langlands correspondence

#Contents#
* table of contents
{:toc}

## Idea

The $p$-adic local Langlands correspondence is a version of the [[local Langlands correspondence]] where the representations are over a $p$-adic field, i.e. on one side we have certain $E$-representations of $G(F)$ (for $G$ some reductive group), and on the other we have certain $E$-representations of $\mathrm{Gal}(\overline{F}/F)$, where $E$ and $F$ are both finite extensions of $\mathbb{Q}_{p}$. It should be compatible with the [[mod p local Langlands correspondence]]. The only case that is known is when $G=\mathrm{GL}_{2}$ and $F=\mathbb{Q}_{p}$.

An approach to realizing one direction of the p-adic local Langlands correspondence using patching was developed in [#CEGGPS2013](#CEG+2013). An approach going in the other direction was developed in [#Scholze2015](#Scholze2015). These two constructions are compatible with each other. Scholze's approach was further developed in [#HansenMann2022](#HansenMann2022).

## Scholze's p-adic Jacquet-Langlands functor

We describe the construction in [#Scholze2015](#Scholze2015). Let $F$ be a finite extension of $\mathbb{Q}_{p}$ and let $C=\widehat{\overline{F}}$. Let $\mathcal{M}_{\infty}$ be the infinite level [[Lubin-Tate space]] over $C$. It is a perfectoid space equipped with commuting actions of $\GL_{n}(F)$ and $D^{\times}$, where $D/F$ is the central simple algebra of invariant $1/n$. There is a Gross-Hopkins period map $\pi_{\mathrm{GH}}:\mathcal{M}_{\infty}\to\mathbb{P}_{C}^{n-1}$.

Let $\pi$ be a smooth $\mathbb{F}_{p}$-representation of $\GL_{n}(F)$. We define a sheaf $\mathcal{F}_{\pi}$ on $\mathbb{P}_{C}^{n-1}$ as follows. Given an etale map $U\to\mathbb{P}_{C}^{n-1}$, define

$$\mathcal{F}_{\pi}=\mathrm{Map}_{\mathrm{cont},\GL_{n}(F)\times D^{\times}}(\vert U\times_{\mathbb{P}_{C}^{n-1}} \mathcal{M}_{\infty}\vert,\pi).$$

Scholze's p-adic Jacquet-Langlands functor $\mathcal{J}^{i}$ is now given by the $i$-th etale cohomology of the sheaf $\mathcal{F}_{\pi}$:

$$\mathcal{J}^{i}(\pi)=H^{i}(\mathbb{P}_{C}^{n-1},\mathcal{F}_{\pi})$$

The representations $\mathcal{J}^{i}(\pi)$ are smooth $\mathbb{F}_{p}$-representations of $D^{\times}$ and also carry an action of the Weil group $W_{F}$.

## References

* {#Breuil2010} [[Christophe Breuil]], *The emerging p-adic Langlands program*, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010 ([pdf](https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/ICM2010.pdf))

* {#CEGGPS2013} Ana Caraiani, [[Matthew Emerton]], [[Toby Gee]], David Geraghty, Vytautas Paskunas, Sug Woo Shin, *Patching and the p-adic local Langlands correspondence* ([arXiv:1310.0831](https://arxiv.org/abs/1310.0831))

* {#Scholze2015} [[Peter Scholze]], *On the p-adic cohomology of the Lubin-Tate tower* ([arXiv:1506.04022](https://arxiv.org/abs/1506.04022))

* {#HansenMann2022} David Hansen, [[Lucas Mann]], *$p$-adic sheaves on classifying stacks, and the $p$-adic Jacquet-Langlands correspondence* ([arXiv:2207.04073](https://arxiv.org/abs/2207.04073))


p-adic modular form

# Contents
* table of contents
{: toc}

## Idea

A _p-adic modular form_ is a limit of [[modular forms]] under a topology that encodes congruences between its Fourier coefficients.

## Definition

The original definition is due to Serre. Consider the space of [[modular forms]] over $\mathbb{Q}$, and recall that they are fully determined by their q-expansions, therefore we may write them as [[power series]] $f=\sum_{n=0}^{\infty}{a_{n}}q^{n}\in\mathbb{Q}[[q]]$. Since $\mathbb{Q}$ embeds into the [[p-adic numbers ]] $\mathbb{Q}_{p}$, we may consider these modular forms as elements of $\mathbb{Q}_{p}[[q]]$ as well. We define a [[valuation]] $v_{p}$ on this space by

$$v_{p}(f):=\inf_{n}( v_{p}(a_{n}))$$

where the $v_{p}$ on the right-hand side is the p-adic valuation on $\mathbb{Q}_{p}$ with $v_{p}(p)=1$.

A _p-adic modular form_ is a power series $f=\sum_{n=0}^{\infty}{a_{n}}q^{n}\in\mathbb{Q}_{p}[[q]]$ such that there exists a sequence of modular forms $f_{1},f_{2},\ldots$ with $f_{i}\to f$.

Another different definition was later formulated by Katz.

## Over rigid analytic spaces

\begin{definition}([Calegari13](#Calegari13), Definition 2.1.13)

Let $X$ be a [[modular curve]] and let $X^{\rig}$ be its associated [[rigid analytic space]]. The p-adic modular forms of weight $k$ are the global sections $H^{0}(X^{\rig}[0],\omega^{k})$, where $X^{\rig}[0]$ denotes the ordinary locus of $X^{\rig}$.

\end{definition}

## Related concepts

* [[modular form]]

* [[eigenvariety]]

* [[Hida theory]]

## References

* {#Calegari13} Frank Calegari, _Congruences Between Modular Forms_, 2013 ([pdf](https://swc-math.github.io/aws/2013/2013CalegariLectureNotes.pdf))

* {#Eischen21} Ellen Eischen, _An Introduction to Eisenstein Measures_, 2021 ([arxiv:2101.01879](https://arxiv.org/abs/2101.01879))


p-adic number

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

# $p$-adic numbers
* table of contents
{: toc}

## Idea

For $p$ any [[prime number]], the _$p$-adic [[numbers]]_ (or _$p$-adic rational numbers_, for emphasis) form a [[field]] $\mathbb{Q}_p$ that [[complete field|completes]] the field of [[rational numbers]] with respect to a [[metric]], called the $p$-adic metric. As such they are analogous to [[real numbers]]. A crucial difference is that the reals form an [[archimedean field]], while the $p$-adic numbers form a [[non-archimedean field]].

Fields of $p$-adic numbers play a role in non-archimedean [[analytic geometry]] that is analogous to the role of the [[real numbers]]/[[Cartesian spaces]] in ordinary [[differential geometry]].

Moreover, as such they serve as an approximation technique in [[arithmetic geometry]] over [[prime fields]] $\mathbb{F}_p$ (see e.g. [Lubicz](#Lubicz)).

There have long been speculations (see the references [below](#ReferencesApplications)) that this must mean that $p$-adic numbers also play a central role in the description of [[physics]], see _[[p-adic physics]]_.

## Definition

We first recall the definition and construction of the [[p-adic integers]]

* [Recollection of the p-adic integers](#RecollectionOfPAdicIntegers)

and then consider

* [The p-adic numbers proper](#PAdicNumbersProper)

### Recollection of the $p$-adic integers
 {#RecollectionOfPAdicIntegers}

Let $\mathbf{Z}$ be the [[ring]] of [[integers]] and for every $q\neq 0$, $q\mathbf{Z}$ its ideal consisting of all integer multiples of $q$, and $\mathbf{Z}/q\mathbf{Z}$ the corresponding [[quotient]], the ring of residues mod $q$.

Let now $p\in \mathbf{Z}_+$ be a [[prime number]]. Then for any two positive integers $n\geq m$ there is an inclusion $p^n \mathbf{Z}\subset p^m\mathbf{Z}$ which induces the canonical homomorphism of quotients $\phi_{n,m}:\mathbf{Z}/p^n\mathbf{Z}\to \mathbf{Z}/p^m\mathbf{Z}$. These homomorphisms for all pairs $n\geq m$ form a family closed under composition, and in fact a category, which is in fact a poset, and moreover a directed system of (commutative unital) rings. The __ring of [[p-adic integers]]__ $\mathbf{Z}_p$ is the (inverse) [[limit]] of this directed system (in the [[category]] [[Ring]] of [[rings]]).

Regarding that the rings in the system are finite, it is clear that the underlying set of $\mathbf{Z}_p$ has a natural topology as a [[profinite space|profinite]] ([[Stone space|Stone]]) space and it is in particular a [[compact space|compact]] [[Hausdorff topological space|Hausdorff topological ring]]. More concretely, $\mathbf{Z}_p$ is the closed (hence compact) subspace of the cartesian product $\prod_{n} \mathbf{Z}/p^n\mathbf{Z}$ of discrete topological spaces $\mathbf{Z}/p^n\mathbf{Z}$ (which is by the [[Tihonov theorem]] compact Hausdorff) consisting of [[thread]]s, i.e. sequences of the form $x = (...,x_n,...,x_2,x_1)$ with $x_n\in p^n\mathbf{Z}$ and satisfying $\phi_{n,m}(x_n) = x_m$.

The kernel of the projection $pr_n: \mathbf{Z}_p\to\mathbf{Z}/p^n\mathbf{Z}$, $x\mapsto x_n$ to the $n$-th component (which is the corresponding projection of the limiting cone) is $p^n\mathbf{Z}_p\subset\mathbf{Z}_p$, i.e. the sequence

$$
0 \longrightarrow \mathbf{Z}_p\stackrel{p^n}\longrightarrow \mathbf{Z}_p\longrightarrow \mathbf{Z}/p^n\mathbf{Z} \longrightarrow 0
$$

is an [[exact sequence]] of [[abelian groups]], hence also $\mathbf{Z}_p/p^n\mathbf{Z}_p\cong \mathbf{Z}/p^n\mathbf{Z}$.

An element $u$ in $\mathbf{Z}_p$ is invertible (and called a $p$-adic unit) iff $u$ is not divisible by $p$.

Let $U\subset\mathbf{Z}_p$ be the group of all invertible elements in $\mathbf{Z}_p$. Then _every element $x\in \mathbf{Z}_p$ can be uniquely written as $s= u p^n$ with $n\geq 0$ and $u\in U$_. The correspondence $x\mapsto n$ defines a [[discrete valuation]] $v_p:\mathbf{Z}_p\to \mathbf{Z}\cup\{\infty\}$ called the  [[p-adic valuation]] and $n$ is said to be the $p$-adic valuation of $x$. Of course, $v_p(0)=\infty$ as required by the axioms of valuation. The [[norm]] induced by the valuation is (up to equivalence) given by ${|x|}_p = p^{-v_p(x)}$, and this in turn induces a metric

$$
d(x,y) = {|x-y|}_p,
$$

making the ring $\mathbf{Z}_p$ a [[complete metric space]] and in fact a completion of $\mathbf{Z}$, in that $d$ is a complete [[metric]], and $\mathbf{Z}$ is dense in it.

Concretely, a $p$-adic integer $x$ may be written as a base-$p$ expansion

$$x = \sum_{n \geq 0} a_n p^n$$

with $a_n \in \{0, 1, \ldots, p-1\}$. Addition and multiplication are performed with [[carrying]] as in ordinary base-$p$ arithmetic, carried infinitely far to the left if $x$ is written as $\ldots a_n a_{n-1} \ldots a_1 a_0$.

#### As an endomorphism ring

Algebraically, the ring of $p$-adic integers is isomorphic to the endomorphism ring $\hom(\mathbf{Z}(p^\infty), \mathbf{Z}(p^\infty))$ where $\mathbf{Z}(p^\infty)$ is the [[Pruefer group|Prüfer group]] $\mathbf{Z}(p^\infty) \coloneqq \mathbb{Z}[1/p]/\mathbb{Z}$. In particular, $\mathbf{Z}(p^\infty)$ is tautologically a $\mathbf{Z}_p$-module.

Relatedly, the additive group of $p$-adic integers is [[Pontrjagin duality|Pontrjagin dual]] to $\mathbf{Z}(p^\infty)$. Observe that $\mathbf{Z}(p^\infty)$ embeds in $S^1$ as the set of all [[root of unity|roots of unity]] of order $p^n$, and that every character $\chi: \mathbf{Z}(p^\infty) \to S^1$ factors through this embedding $\mathbf{Z}(p^\infty) \hookrightarrow S^1$.

### The $p$-adic numbers proper
 {#PAdicNumbersProper}

The __field of $p$-adic numbers__ $\mathbf{Q}_p$ is the [[field of fractions]] of the [[p-adic integers]] $\mathbf{Z}_p$. The $p$-adic valuation $v_p$ extends to a discrete valuation, also denoted $v_p$ on $\mathbf{Q}_p$. Indeed, it is still true for all $x\in \mathbf{Q}_p$ that they can be uniquely written in the form $p^n u$ where $u\in U$ (the same group $U$ as before), but now one needs to allow $n\in \mathbf{Z}$. One defines the metric on $\mathbf{Q}_p$ by the same formula as for $\mathbf{Z}_p$. It appears that $\mathbf{Q}_p$ is a [[complete field]] (in particular locally compact Hausdorff) and that $\mathbf{Z}_p$ is an *open* subring.

The distance $d$ satisfies the "ultrametric" inequality

$$
d(x,z) \leq sup\{d(x,y),d(y,z)\}
$$

Concretely, a $p$-adic number $x$ may be written as $\sum_{n \geq k} a_n p^n$, with only finitely many negative powers of $p$ occurring. If $k \lt 0$, the expansion is conventionally displayed as

$$x = \ldots a_1 a_0.a_{-1} \ldots a_k$$

with finitely many terms to the "right" of the "decimal" point. Again such expressions are added and multiplied with carrying as in ordinary arithmetic.

## Properties

### Basic properties

+-- {: .num_prop #InvertiblepAdic}
###### Proposition

An element $x\in \mathbb{Z}_p$ is invertible precisely if $x_0 \neq 0$.

=--

### As a field completion

[[Ostrowski's theorem]] implies there are precisely two kinds of [[complete field|completions]] of the [[rational numbers]]: the [[real numbers]] and the $p$-adic numbers.

+-- {: .num_theorem}
###### Theorem
**([[Ostrowski's theorem]])**

Any non-trivial [[absolute value]] on the [[rational numbers]] is equivalent either to the standard real absolute value, or to the $p$-adic absolute value.

=--

### Topological disconnectedness and G-topology
 {#Disconnectedness}

While the $p$-adic numbers are complete in the [[p-adic norm]], that [[topology]] is exotic: $\mathbb{Q}_p$ is a locally compact, Hausdorff, [[totally disconnected topological space]].

For that reason the naive idea of formulating [[p-adic geometry]] in analogy to [[complex analytic geometry]] as modeled on domains in $\mathbb{Q}_p^n$, regarded with their [[subspace topology]], fails (for instance there would be no [[analytic continuation]]), as also all these domains are totally disconnected.

Instead there is ([Tate 71](#Tate71)) a suitable [[Grothendieck topology]] on such [[affinoid domains]] -- the _[[G-topology]]_ -- with respect to which there is a good theory of [[non-archimedean analytic geometry]] ("[[rigid analytic geometry]]") and hence in particular of [[p-adic geometry]]. Moreover, one may sensibly assign to an $p$-adic domain a [[topological space]] which _is_ well behaved (in particular locally connected and even locally contractible), this is the _[[analytic spectrum]]_ construction. The resulting topological space is equipped with covers by [[affinoid domain]] under the [[analytic spectrum]] are called [[Berkovich spaces]].

### Pontryagin duality

Earlier we observed that as an additive compact Hausdorff [[topological group]], the inverse limit $\mathbf{Z}_p = \lim_{\leftarrow n} \mathbb{Z}/(p^n)$ is [[Pontryagin duality|dual]] to the discrete [[Pruefer group|Prüfer group]] $\mathbf{Z}(p^\infty) \coloneqq \mathbb{Z}[1/p]/\mathbb{Z}$ that is isomorphic to a direct limit of finite cyclic groups $\lim_{\to n} \mathbb{Z}/(p^n)$. The canonical inclusion $\mathbb{Z}[1/p] \to \mathbf{Q}_p$ induces an isomorphism $\mathbf{Z}(p^\infty) \to \mathbf{Q}_p/\mathbf{Z}_p$, in fact an isomorphism of $\mathbf{Z}_p$-modules, so there is an exact sequence

$$0 \to \mathbf{Z}_p \stackrel{i}{\hookrightarrow} \mathbf{Q}_p \stackrel{q}{\to} \mathbf{Z}(p^\infty) \to 0.$$

This exact sequence is Pontrjagin self-dual in the sense that the map $\mathbf{Q}_p \to \mathbf{Q}_p^\wedge$ induced from the pairing

$$\mathbf{Q}_p \times \mathbf{Q}_p \stackrel{mult}{\to} \mathbf{Q}_p \stackrel{q}{\to} \mathbb{Z}[1/p]/\mathbb{Z} \hookrightarrow \mathbb{R}/\mathbb{Z}$$

fits into an isomorphism of exact sequences

$$\array{
0 & \to & \mathbf{Z}_p & \stackrel{i}{\to} & \mathbf{Q}_p & \stackrel{q}{\to} & \mathbf{Z}(p^\infty) & \to & 0 \\
 & & \downarrow & & \downarrow & & \downarrow & & \\
0 & \to & \mathbf{Z}(p^\infty)^\wedge & \stackrel{q^\wedge}{\to} & \mathbf{Q}_p^\wedge & \stackrel{i^\wedge}{\to} & \mathbf{Z}_p^\wedge & \to & 0
}$$

where the commutativity of the squares can be traced to the fact that $q$ is a $\mathbf{Z}_p$-module homomorphism, and where the vertical isomorphisms on left and right come from [[Pontrjagin duality]]. The middle arrow is then an isomorphism by the [[short five lemma]] for [[topological groups]], which holds by protomodularity of topological groups.

This self-duality figures in Tate's thesis; for more, see [[ring of adeles]].

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[ring of adeles]]

* [[carrying]]

* [[p-adic integration]]

* [[natural number]], [[integer]], [[rational number]], [[algebraic number]], [[Gaussian number]], [[irrational number]], [[real number]]

* [[p-adic integer]], [[p-adic complex number]]

* [[p-adic geometry]]

  * [[p-adic analytic space]], [[non-commutative analytic space]]

* [[p-adic homotopy]]

* [[p-adic cohomology]]

* [[p-adic physics]]

  * [[p-adic string theory]]

  * [[p-adic AdS/CFT correspondence]]

## References
 {#References}

The $p$-adic numbers had been introduced in

* [[Kurt Hensel]], _Über eine neue Begründung der Theorie der algebraischen Zahlen_ Jahresbericht der Deutschen Mathematiker-Vereinigung 6 (3): 83–88. (1897)

A standard reference is

* [[Jean-Pierre Serre]], _A course in arithmetic_, Grad. Texts in Math. __7__, Springer (1973)

Review in the context of [[p-localization|p-local]] [[homotopy theory]] is in

* {#Sullivan70} [[Dennis Sullivan]], pp. 9 of _Localization, Periodicity and Galois Symmetry_ (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer ([pdf](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf))

Review of the use of $p$-adic numbers in [[arithmetic geometry]] includes

* {#Lubicz} [[David Lubicz]], _An introduction to the algorithmic of $p$-adic numbers_ ([pdf](http://www.hyperelliptic.org/tanja/conf/summerschool08/slides/p-adics.pdf))

A formalization in [[homotopy type theory]] and there in [[Coq]] is discussed in

* {#PelayoVoevodskyWarren13} Álvaro Pelayo, [[Vladimir Voevodsky]], [[Michael Warren]], _A preliminary univalent formalization of the p-adic numbers_ ([arXiv:1302.1207](http://arxiv.org/abs/1302.1207))

$p$-adic [[differential equations]] are discussed in

* [[Kiran Kedlaya]], _$p$-adic differential equations_ ([pdf](http://math.ucsd.edu/~kedlaya/18.787/compiled.pdf), [course notes](http://math.ucsd.edu/~kedlaya/18.787/))

* [[Gilles Cristol]], _Exposants $p$-adiques et solutions dans les couronnes_ ([pdf](http://www.math.jussieu.fr/~christol/exposants.pdf))

The development of [[rigid analytic geometry]] starts with

* {#Tate71} [[John Tate]], _Rigid analytic spaces_, Invent. Math. __12__:257–289, 1971.

[[p-adic homotopy theory]] is discussed in

* [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

[[!redirects p-adic]]

[[!redirects p-adic number]]
[[!redirects p-adic numbers]]

[[!redirects adic number]]
[[!redirects adic numbers]]

[[!redirects p-adic number field]]
[[!redirects p-adic number fields]]
[[!redirects p-adic field]]
[[!redirects p-adic fields]]

[[!redirects p-adic valuation]]
[[!redirects p-adic valuations]]

[[!redirects p-adic norm]]
[[!redirects p-adic norms]]

[[!redirects p-adic rational number]]
[[!redirects p-adic rational numbers]]

[[!redirects ring of p-adic numbers]]
[[!redirects rings of p-adic numbers]]

[[!redirects ring of p-adic rational numbers]]
[[!redirects rings of p-adic rational numbers]]


p-adic physics

+-- {: .rightHandSide}
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are various attempts/speculations/proposals to use [[p-adic numbers]] in place of [[real numbers]] in the description of [[physics]].

Historically what initiated much of this development was the observation in ([Volovich 87](#Volovich87), reviewed in [VVZ 95, section XIV](#VVZ95)) that the [[integral]] expression for the [[Veneziano amplitude]] of the [[open string|open]] [[bosonic string]] naturally generalizes from an integral over the [[real numbers]] (which in this case parameterize the [[boundary]] of the [[open string]] [[worldsheet]]) to the [[p-adic numbers]]. Since this concerns the [[bosonic string]] [[tachyon]] [[state]], such [[p-adic string theory]] has been discussed a lot in the context of [[tachyon condensation]] and [[Sen's conjecture]] ([Cottrell 02](#Cottrell02)).

Generally, the development of string theory has shown that its [[worldsheet]] is usefully regarded as an object in [[algebraic geometry]] (see also at _[[number theory and physics]]_) and mathematically the generalization from [[algebraic varieties]] over the [[complex numbers]] to more general algebraic varieties (or [[schemes]]) is often natural, if not compelling. For instance when the [[Witten genus]] (essentially the [[partition function]] of the [[superstring]]) is refined to the [[string orientation of tmf]] then the [[elliptic curves]] over the [[complex numbers]] which serve as the [[torus|toroidal]] [[worldsheets]] over the [[complex numbers]] are generalized to [[elliptic curves]] over general [[rings]] and by the [[fracture theorems]] the computations in [[tmf]] in fact typically proceed by decomposing the general problem into that of ellitpic curves over the [[rational numbers]] and over the [[p-adic integers]]. See at _[[p-adic string theory]]_ for more on this.

## Related entries

* [[number theory and physics]]

* [[p-adic string theory]]

* [[p-adic AdS/CFT correspondence]]

* [[arithmetic Chern-Simons theory]]

* [[p-adic geometry]]

* [[p-adic homotopy]]

* [[p-adic cohomology]]

## References

### General

Textbooks:

* {#VVZ95} Vasiliĭ Vladimirov, I. V. Volovich, E. I. Zelenov, _$p$-Adic analysis and mathematical physics_, World Scientific, 1995

Reviews:

* L. Brekke, P. Freund, $p$-adic numbers in physics, Phys. Rep. 233 (1993)

* [p-adic and adelic physics website](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics7.htm)

* Branko Dragovich, _Non-Archimedean Geometry and Physics on Adelic Spaces_ ([arXiv:math-ph/0306023](http://arxiv.org/abs/math-ph/0306023))

* Branko Dragovich, _$p$-Adic and Adelic Quantum Mechanics_ ([arXiv:hep-th/0312046](http://arxiv.org/abs/hep-th/0312046))

* B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, _$p$-Adic Mathematical Physics_ ([arXiv:0904.4205](http://arxiv.org/abs/0904.4205))

* B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov, _$p$-Adic Mathematical Physics: The First 30 Years_ ([arXiv:1705.04758](https://arxiv.org/abs/1705.04758))

Further development

* An Huang, Dan Mao, [[Bogdan Stoica]], _From p-adic to Archimedean Physics: Renormalization Group Flow and Berkovich Spaces_ ([arXiv:2001.01725](https://arxiv.org/abs/2001.01725))

* Dmitry S. Ageev, Andrey A. Bagrov, Askar A. Iliasov, _Coleman-Weinberg potential in p-adic field theory_ ([arXiv:2004.03014](https://arxiv.org/abs/2004.03014))

### String theory

Discussion of the [[Veneziano amplitude]] for [[p-adic string theory]] is originally due to

* {#Volovich87} I. V. Volovich, _p-Адическое пространство-время и теория струн_, ТМФ, 71:3 (1987)[free Rus. pdf](http://www.mathnet.ru/links/704079ec1d28d91ba6eb6d492359b387/tmf4962.pdf); transl. _$p$-adic space-time and string theory_, Theor. Math. Phys. __71__, 574–576 (1987), [eng doi](http://dx.doi.org/10.1007/BF01017088), [nonfree Eng. pdf](http://www.springerlink.com/content/k514t553607324n0/fulltext.pdf)

See also

* {#Cottrell02} William Cottrell, _$p$-adic Strings and Tachyon Condensation_, 2002 ([pdf](http://jfi.uchicago.edu/~tten/teaching/Phys.291/Cottrell_Freund_2002.pdf))

* Branko Dragovich, _Nonlocal dynamics of p-adic strings_, ([arxiv/1011.0912](http://arxiv.org/abs/1011.0912))

A relation of $p$-adic string theory to number-theoretic [[Langlands duality]] is hypothesized and explored somewhat in

* Michele Nardelli, _On the link between the structure of A-branes observed in the homological mirror symmetry and the classical theory of automorphic forms: mathematical connections with the modular elliptic curves, $p$-adic and adelic numbers and $p$-adic and adelic strings_, in [p-adic and adelic physics](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics7.htm) ([pdf](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli_branes.pdf))


p-adic string theory

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* table of contents
{:toc}

## Idea

What is called _$p$-adic string theory_ is the study of a variant of [[scattering amplitudes]] in [[string theory]] (hence [[string scattering amplitudes]]) where the [[worldsheet]] of the [[string]] is regarded not as a [[Riemann surface]] but as an object in [[p-adic geometry]].

### Open string amplitudes

The original observation is that of ([Volovich 87](#Volovich87), reviewed in [VVZ 95, section XIV](#VVZ95)) that the [[integral]] expression for the [[Veneziano amplitude]] of the [[open string|open]] [[bosonic string]] naturally generalizes from an integral over the [[real numbers]] (which in this case parameterize the [[boundary]] of the [[open string]] [[worldsheet]]) to the [[p-adic numbers]] by _[[adelic integration]]_.

The standard [[Veneziano amplitude]] has the expression

$$
  \int_{\mathbb{R}}
  {\vert x\vert}^{a-1}
  \cdot
  {\vert 1-x\vert}^{b-1}
  d x
$$

where the [[norm]] involved is the usual [[absolute value]], and the proposed $p$-adic version is hence

$$
  \int_{\mathbb{Q}_p}
  {\vert x\vert}_p^{a-1}
  \cdot
  {\vert 1-x\vert}_p^{b-1}
  d x
$$

with the [[p-adic norm]] instead.

The main result here is ([Freund-Witten 87](#FreundWitten87)) that the ordinary [[Veneziano amplitude]] equals the inverse of the product of its $p$-adic versions, for all primes $p$, apparently a version of the [idelic product formula](group%20of%20ideles#ProductFormula).

With due regularization this result carries over to other [[string scattering amplitudes]], too. When forming these products one also speaks of _[[ring of adeles|adelic]] string theory_.

Since the [[Veneziano amplitude]] concerns the [[bosonic string]] [[tachyon]] [[state]], [[p-adic string theory]] has been discussed a lot in the context of [[tachyon condensation]] and [[Sen's conjecture]] ([Cottrell 02](#Cottrell02)).

Traditionally literature on $p$-adic string theory asserts that the generalization of this from the [[open string]] to the [[closed string]] remains unclear (e.g [CMZ 89, section 4](#CMZ89), [Cottrell 02, section 5](#Cottrell02)), since it is unclear which adic version of the [[complex numbers]] to use. However, in other parts of the literature adic versions of closed strings are common, this we discuss [below](#TopologicalWittenGenus).

### Closed string 1-loop vacuum amplitudes (topological Witten genus)
 {#TopologicalWittenGenus}

Generally, the development of string theory has shown that its [[worldsheet]] is usefully regarded as an object in [[algebraic geometry]] (see also at _[[number theory and physics]]_) and mathematically the generalization from [[algebraic varieties]] over the [[complex numbers]] to more general algebraic varieties (or [[schemes]]) is often natural, if not compelling.

For instance when the [[Witten genus]] (essentially the [[partition function]] of the [[superstring]]) is refined to the [[string orientation of tmf]] then the [[elliptic curves]] over the [[complex numbers]] which serve as the [[torus|toroidal]] [[worldsheets]] over the [[complex numbers]] are generalized to [[elliptic curves]] over general [[rings]] and by the [[fracture theorems]] the computations in [[tmf]] in fact typically proceed (see [here](tmf#DecomopositionViaArithmeticSquares)) by decomposing the general problem into that of elliptic curves over the [[rational numbers]] and over the [[p-adic integers]]. The result refines the [[Witten genus]]

$$
  \Omega^String_{\bullet} \longrightarrow MF_\bullet
$$

(being a [[ring]] [[homomorphism]]) from the [[String structure]] [[cobordism ring]] to that of [[modular forms]] to one of [[E-∞ rings]]

$$
  M String \longrightarrow tmf
$$

from the [[String structure]] [[Thom spectrum]] to [[tmf]]. Notice that $M String$ here classifies String-[[cobordism cohomology theory|cobordism]] and hence parameterizes ordinary (not $p$-adic) [[target space|target]] [[spacetime]] [[manifolds]], while $tmf$ on the right does regard the [[genus of a surface|genus]]-1 [[worldsheet]] as a general [[elliptic curve]], hence in particular possibly as an [elliptic curve over the p-adic integers](elliptic%20curve#OverpAdics).

## Related concepts

* The interesting aspects of $p$-adic string theory have led people to consider _[[p-adic physics]]_ more generally. But it remains noteworthy that in $p$-adic string theory it is exactly only the [[worldsheet]] which is regarded in [[p-adic geometry]], while for instance the [[complex numbers]] as they appear as [[coefficients]] of [[quantum physics]] are not replaced by $p$-adics.

* [[p-adic AdS-CFT]]

* [[number theory and physics]]

* [[p-adic geometry]]

* [[p-adic homotopy]]

* [[p-adic cohomology]]

## References

The original articles include

* [[Peter Freund]], Mark Olson, _Non-archimedean strings_, Physics Letters B 199,2 (1987) (arXiv:10.1016/0370-2693(87)91356-6)

* {#Volovich87} I. V. Volovich, _p-Адическое пространство-время и теория струн_, ТМФ, 71:3 (1987)[free Rus. pdf](http://www.mathnet.ru/links/704079ec1d28d91ba6eb6d492359b387/tmf4962.pdf); transl. _$p$-adic space-time and string theory_, Theor. Math. Phys. __71__, 574–576 (1987), [eng doi](http://dx.doi.org/10.1007/BF01017088), [nonfree Eng. pdf](http://www.springerlink.com/content/k514t553607324n0/fulltext.pdf)

The early history of the subject is recalled in

* Paul H. Frampton, _Particle Theory at Chicago in Late Sixties and p-Adic Strings_ ([arXiv:2001.10915](https://arxiv.org/abs/2001.10915))

That the ordinary [[Veneziano amplitude]] is the inverse product of all its $p$-adic versions is due to

* {#FreundWitten87} [[Peter Freund]], [[Edward Witten]], _Adelic string amplitudes_, Phys. Lett. B 199, 191 (1987). ([web record](http://adsabs.harvard.edu/abs/1987PhLB..199..191F))

also

* Lee Brekke,  [[Peter Freund]],  Mark Olson, [[Edward Witten]], _Non archimedean  string dynamics_, Nucl. Phys. B302, 3 (1988) (arXiv:10.1016/0550-3213(88)90207-6)

A detailed discussion of $p$-adic open [[string scattering amplitudes]] is in

* {#CMZ89} L. Chekhov, A. Mironov, A. Zabrodin, _Multiloop calculations in $p$-adic String theory and Bruhat-Tits Trees_, Comm. Math. Phys. 125, 675-711 (1989) ([Euclid](http://projecteuclid.org/euclid.cmp/1104179635))

A review of this is in

* Lee Brekke and [[Peter Freund]], _$p$-Adic numbers in physics_, Phys. Rep. 233, 1 (1993) ([web record](http://adsabs.harvard.edu/abs/1993PhR...233....1B))

and with an eye towards [[AdS-CFT duality]] in

* [[Peter Freund]], _$p$-adic Strings Then and Now_ ([arXiv:1711.00523](https://arxiv.org/abs/1711.00523))

Suggestion that the [[disk]] [[worldsheet]] of the [[open string|open]] [[p-adic string]] is to be identified with the [[Bruhat-Tits tree]] $T_p$:

* [[Anton Zabrodin]], _Non-Archimedean strings and Bruhat-Tits trees_, Comm. Math. Phys.
Volume 123, Number 3 (1989), 463-483 ([euclid.cmp/1104178891](https://projecteuclid.org/euclid.cmp/1104178891))

Discussion of [[tachyon condensation]] in $p$-adic string theory includes

* {#Cottrell02} William Cottrell, _$p$-adic Strings and Tachyon Condensation_, 2002 ([pdf](http://jfi.uchicago.edu/~tten/teaching/Phys.291/Cottrell_Freund_2002.pdf))

See also

* {#BGZ16} Miriam Bocardo-Gaspar, H. García-Compeán, W. A. Zúñiga-Galindo, _Regularization of p-adic String Amplitudes, and Multivariate Local Zeta Functions_ ([arXiv:1611.03807](https://arxiv.org/abs/1611.03807))

Relation to [[gravity]] and the zeros of the [[Riemann zeta function]] (hence the [[Riemann hypothesis]]):

* {#HuangStoicaYau19} [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013), [spire:1712659](http://inspirehep.net/record/1712659/))

Discussion of [[BTZ black holes]]:

* [[Ingmar Saberi]], around Figure 4.3 of: _Knots, Trees, and Fields: Common Ground Between Physics and Mathematics_, 2016 ([doi:10.7907/Z9VX0DHZ](https://thesis.library.caltech.edu/9896), [pdf](https://thesis.library.caltech.edu/9896/1/Saberi_Ingmar_2016.pdf))

Proposal for generalization to $p$-adic [[superstrings]]:

* Hugo Garcia-Compean, Edgar Y. Lopez, *Towards Non-archimedean Superstrings* [[arXiv:2202.02425](https://arxiv.org/abs/2202.02425)]

See also:

* Christian B. Jepsen, *Adelic Amplitudes and Intricacies of Infinite Products* [[arXiv:2211.01611](https://arxiv.org/abs/2211.01611)]

[[!redirects p-adic string theories]]

[[!redirects adelic string theory]]
[[!redirects adelic string theories]]

[[!redirects p-adic string]]
[[!redirects p-adic strings]]


p-adic Teichmüller theory

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\tableofcontents

## Idea ##

A version of [[Teichmüller theory]] for [[p-adic numbers]] rather than the [[complex numbers]].

## See also ##

* [[Teichmüller theory]]

## References ##

* [[Kirti Joshi]], *Construction of Arithmetic Teichmüller Spaces and some applications* ([arXiv:2106.11452](https://arxiv.org/abs/2106.11452))

* [[Kirti Joshi]], *Construction of Arithmetic Teichmüller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12* ([arXiv:2303.01662](https://arxiv.org/abs/2303.01662))

* [[Kirti Joshi]], *Construction of Arithmetic Teichmüller spaces II($\frac{1}{2}$): Deformations of Number Fields* ([arXiv:2305.10398](https://arxiv.org/abs/2305.10398))

* [[Kirti Joshi]], *Comments on Arithmetic Teichmüller Spaces* ([arXiv:2111.06771](https://arxiv.org/abs/2111.06771))

* [[Kirti Joshi]], *Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups -- Arithmetic Holomorphic Structures* ([arXiv:2210.11635](https://arxiv.org/abs/2210.11635))

[[!redirects arithmetic Teichmüller theory]]

[[!redirects arithmetic Teichmüller space]]
[[!redirects arithmetic Teichmüller spaces]]

[[!redirects p-adic Teichmüller space]]
[[!redirects p-adic Teichmüller spaces]]


p-compact group

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## Idea

The _$p$-compact groups_ ([Dwyer-Wilkerson 94](#DwyerWilkerson94)) for some [[prime number]] $p$ are a class of [[∞-groups]] that shares crucial properties with the class of [[compact Lie groups]], without actually being compact Lie groups. Hence they are also called _homotopy Lie groups_ ([Møller 95](#Moller95)).

## Properties

### Classification

The classification of $p$-compact groups states that there is a [[bijection]] between [[isomorphism classes]] of [[connected topological space|connected]] [[p-compact groups]], and [[isomorphism classes]] of [[root]] data over the [[p-adic integers]] (as conjectured by [[Clarence Wilkerson]] and others, in various forms, since the early days of the theory).

This is completely analogous to the classification of [[connected topological space|connected]] [[compact Lie groups]], under replacing the [[integers]] $\mathbb{Z}$ by the [[p-adic integers]] $\mathbb{Z}_p$.

Specializing to $p=2$ one gets as a corollary that any [[classifying space]] $B X$ of a connected [[2-compact group]] $X$ splits as

$$
  B X \cong B G \times \big(B DI(4)\big)^s
$$

the [[Cartesian product]] of the [[p-completion|2-completion]] of the [[classifying space]] of the [[compact Lie group]] $G$, and $s$ copies of the [[Dwyer-Wilkerson space]] $B DI(4)$ for some $s$.

$DI(4) = $ [[G3]] corresponds to the finite $\mathbb{Z}_2$-reflection group which is number 24 on the Shepard-Todd list. It is the only irreducible finite complex reflection group which is realizable over $\mathbb{Z}_2$ but not $\mathbb{Z}$.

([Andersen-Grodal 06](#AndersenGrodal06), see [Grodal 10](#Grodal10))

## Examples

* Let $G$ be any compact Lie group whose component group $\pi_0(G)$ is a $p$-group. Define $B \hat{G} = (B G)_p$. Then $\hat{G}$ is a $p$-compact group.

* (Sullivan) The $\mathbb{F}_p$-local sphere $(S^{n-1})_p$, where $n \gt 2$ is an integer dividing $p-1$.

* [[Dwyer-Wilkerson H-space]]

## References

Due to

* {#DwyerWilkerson94} [[William Dwyer]], [[Clarence Wilkerson]], _Homotopy Fixed-point Methods for Lie Groups and Finite Loop Spaces_,  Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 395-442 ([jstor:2946585](https://www.jstor.org/stable/2946585))

Classification:

* {#AndersenGrodal06} Kasper Andersen, [[Jesper Grodal]], _The classification of 2-compact groups_, J. Amer. Math. Soc. 22 (2009), 387-436 ([arXiv:math/0611437](https://arxiv.org/abs/math/0611437))

Review:

* {#Moller95} [[Jesper Møller]], _Homotopy Lie groups_, Bull. Amer. Math. Soc. (N.S.) 32 (1995) 413-428 ([arXiv:math/9510218](https://arxiv.org/abs/math/9510218))

* {#Grodal10} [[Jesper Grodal]], _The Classification of $p$–Compact Groups and Homotopical Group Theory_, Proceedings of the International Congress of Mathematicians, Hyderabad 2010 ([arXiv:1003.4010](https://arxiv.org/abs/1003.4010), [pdf](http://web.math.ku.dk/~jg/papers/icm.pdf), [[GrodalpCompactGroups2010.pdf:file]])

See also

* Wikipedia, _[P-compact group](https://en.wikipedia.org/wiki/P-compact_group)_

[[!redirects p-compact groups]]

[[!redirects homotopy Lie group]]
[[!redirects homotopy Lie groups]]

[[!redirects 2-compact group]]
[[!redirects 2-compact groups]]


p-connection

For a [[smooth morphism]] $p$ of smooth [[analytic space]]s or of [[smooth schemes]] $p\colon X \to S$ a __$p$-connection__ is an $\mathcal{O}_X$-linear map $\nabla_S\colon p^* T_S \to T_X$ such that $\mathrm{d}p \circ \nabla_S = id_{p^* T_S}$. The "[[differential]]" $\mathrm{d}p$ here is the map $T_X \to p^* T_S$ induced by the universality of the [[pullback]] and the differential.  A $p$-connection is __flat/integrable__ if the corresponding ([[adjunct|by adjunction]]) map $T_S \to p_* T_X$ commutes with brackets of [[tangent vector field|vector fields]].

* [[Alexander Beilinson]], [[David Kazhdan]], _Flat projective connections_, <http://www.math.sunysb.edu/~kirillov/manuscripts/kazhdan2.pdf>

[[!redirects p-connection]]
[[!redirects p--connection]]


p-convex polarization

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## Idea

The concept of $p$-convex polarization is the generalization of the concept of [[Kähler polarization]] to possibly indefinite [[Hermitean metrics]].

For a Hermitean metric with precisely $p$ negative eigenvalues, one speaks of _$p$-convex polarization_ (e.g. [Griffiths survey, p. 3](#GriffithsSurvey)).

Hence for $(X,h)$ a [[Hermitean manifold]] with Hermitean metric of [[signature of a quadratic form|signature]] $(dim(X)-p,p)$, then a $p$-convex polarization of it is a [[symplectic form]] $\omega$ which in every local [[coordinate chart]] has the form

$$
  \omega = \tfrac{i}{2} \sum_{\alpha,\beta} h_{\alpha,\beta} d z^\alpha \wedge d\bar z^\beta
$$

where $(h_{\alpha,\beta})$ are the components of the [[Riemannian metric|metric]] in the given chart.

Hence for $p = 0$ then a $p$-convex polarization reduces to a [[Kähler polarization]] or, algebraically, to a [[polarized variety]].

## Examples

The Griffiths [[intermediate Jacobians]] (see there for more) naturally carry a $p$-convex polarization but not in general a [[Kähler polarization]] ([Griffiths 68b](#Griffiths68b)).

## Related concepts

* [[real polarization]]

* [[polarized variety]]

## References

* {#Griffiths68a} [[Phillip Griffiths]], _Periods of integrals on algebraic manifolds. I_ Construction and properties of the modular varieties", American Journal of Mathematics 90 (2): 568–626, (1968) doi:10.2307/2373545, ISSN 0002-9327, JSTOR 2373545, MR 0229641

* {#Griffiths68b} [[Phillip Griffiths]]  _Periods of integrals on algebraic manifolds. II_ Local study of the period mapping", American Journal of Mathematics 90 (3): 805–865 (1968), doi:10.2307/2373485, ISSN 0002-9327, JSTOR 2373485, MR 0233825

* {#GriffithsSurvey} [[Phillip Griffiths]], _On the periods of integrals on algebraic manifolds_, survey ([[GriffithsPeriodsSummary.pdf:file]])

[[!redirects p-convex polarizations]]

[[!redirects convex polarization]]
[[!redirects convex polarizations]]


p-curvature

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## Idea

In [[Riemannian geometry]] the term _$p$-curvature_ ([Labbi 97](#Labbi97)) refers to a generalization of the [[scalar curvature]].

## Related entries

[[!include curvature in Riemannian geometry -- table]]

* [[Stolz conjecture]]

## References

### In Riemannian geometry
 {#InRiemannianGeometry}

The concept was introduced in

* {#Labbi97} [[Mohammed Labbi]], _Stability under surgeries of the $p$-curvature positivity and manifolds with positive Einstein tensor_, Annals of Global Analysis and Geometry, 15 no 4, 299-312 (1997). ([pdf](http://streaming.ictp.trieste.it/preprints/P/96/097.pdf))

Further discussion includes

* [[Mohammed Labbi]], _Manifolds with positive second Gauss-Bonnet curvature_, Pacific Journal of Math. Vol. 227, No. 2, 295-310, (2006) ([pdf](http://msp.org/pjm/2006/227-2/pjm-v227-n2-p04-s.pdf))

* Boris Botvinnik, [[Mohammed Labbi]], _Highly connected manifolds of positive $p$-curvature_, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 ([arXiv:1201.1849](http://arxiv.org/abs/1201.1849), [doi:10.1090/S0002-9947-2014-05939-4](https://doi.org/10.1090/S0002-9947-2014-05939-4))

(see also at [[fivebrane structure]]).

### In algebraic geometry

* Brian Osserman, _Connections, curvature and $p$-curvature_ ([pdf](https://www.math.ucdavis.edu/~osserman/math/connections.pdf))

[[!redirects p-curvatures]]


p-derivation

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##Idea
For a prime $p$, a $p$-derivation on a commutative [[ring]] $R$ is a map $\delta:R \to R$ such that $\phi(r) \coloneqq r^p + p \delta(r)$ is a [[ring homomorphism]]. $\phi$ is called the _Frobenius lift_ attached to $\delta$.

An example of a $p$-derivation is the [[Fermat quotient]].

##Related pages

* [[arithmetic differential geometry]]
* [[Borger's absolute geometry]]
* [[derivation]]
* [[Joyal delta-ring]]

[[!redirects Frobenius lift]]
[[!redirects Frobenius lifts]]
[[!redirects p-derivations]]


p-divisible group

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## Idea

In great generality, for an [[integer]] $p$ a _$p$-divisible group_ is a [[inductive system|codirected diagram]] of [[abelian group|abelian]] [[group objects]] in a [[category]] $C$ where the abelian-group objects are (equivalently) the [[kernel|kernels]] of the map given by multiplication with a power of $p$; these kernels are also called $p^n$-torsions.

In the classically studied case $p$ is a prime number, $C$ is the category of [[scheme|schemes]] over a commutative ring (mostly a field with prime characteristic) and the abelian [[group scheme|group schemes]] occurring in the diagram are assumed to be finite. In this case the diagram defining the $p$-divisible group can be described in terms of the growth of the [[order of a group|order]] of the group schemes in the diagram.

Note that there is also a notion of [[divisible group]].

## Definition

+-- {: .num_defn}
###### Definition

Fix a [[prime number]] $p$, a [[natural number|positive integer]] $h$, and a [[commutative ring]] $R$. Consider the group schemes $G$ over

A **$p$-divisible group of height $h$ over $R$** is a [[codirected limit|codirected diagram]] $(G_\nu, i_\nu)_{\nu \in \mathbb{N}}$ where each $G_\nu$ is a finite commutative [[group scheme]] over $R$ of [[order of a group|order]] $p^{\nu h}$ that also satisfies the property that

$$0\to G_\nu \stackrel{i_\nu}{\to} G_{\nu +1}\stackrel{p^\nu}{\to} G_{\nu +1}$$

is [[exact sequence|exact]]. In other words, the maps of the system identify $G_\nu$ with the [[kernel]] of multiplication by $p^\nu$ in $G_{\nu +1}$.

Some authors refer to $colim_\nu G_\nu$ (instead of the diagram itself) as the $p$-divisible group.
=--

It can be checked that a $p$-divisible group over $R$ is a $p$-torsion commutative [[formal group]] $G$ for which $p\colon G \to G$ is an [[isogeny]].

## Examples

+-- {: .num_example}
###### Example

The kernel of raising to the $p^\nu$ power on $\mathbb{G}_m$ (sometimes called [[p-torsion]]) is a [[group scheme]] $\mu_{p^\nu}$. The limit $\lim_{\to} \mu_{p^\nu}=\mu_{p^\infty}$ is a $p$-divisible group of height $1$.

see Lipnowski [pg.2, example (b)](http://math.stanford.edu/~malipnow/expository/pdiv.pdf)

=--

+-- {: .num_example}
###### Example

The eponymous ($p$-divisible groups are sometimes called Barsotti-Tate groups) example is a special case of the previous one - namely [[the Barsotti-Tate group of an abelian variety]]. Let $X$ be an [[abelian variety]] over $R$ of dimension $g$, then the multiplication map by $p^\nu$ has kernel $_{p^\nu}X$ which is a finite [[group scheme]] over $R$ of order $p^{2g \nu}$. The natural inclusions satisfy the conditions for the limit denoted $X(p)$ to be a $p$-divisible group of height $2g$.

=--

+-- {: .num_example}
###### Example

A theorem of [Serre and Tate](#Tate) says that there is an [[equivalence of categories]] between divisible, commutative, formal [[Lie group]]s over $R$ and the category of connected $p$-divisible groups over $R$ given by $\Gamma \mapsto \Gamma (p)$, where $\Gamma(p)=\lim_{\to} \mathrm{ker}(p^n)$. In particular, every connected $p$-divisible group is smooth {#Tate}

=--

## The Cartier dual

* Given a $p$-divisible group $G$, each individual $G_\nu$ has a [[Cartier dual]] $G_\nu^D$ since they are all group schemes. There are also maps $j_\nu$ that make the composite $G_{\nu+1}\stackrel{j_\nu}{\to} G_\nu \stackrel{i_\nu}{\to} G_{\nu +1}$ the multiplication by $p$ on $G_{\nu +1}$. After taking duals, the composite is still the multiplication by $p$ map on $G_{\nu +1}^D$, so it is easily checked that $(G_{\nu}^D, j_{\nu}^D)$ forms a $p$-divisible group called the Cartier dual.

* One of the important properties of the Cartier dual is that one can determine the height of a $p$-divisible group (often a hard task when in the abstract) using the information of the dimension of the formal group and its dual. For any $p$-divisible group, $G$, we have the formula that $ht(G)=ht(G^D)=\dim G + \dim G^D$.

## Dieudonné modules

For the moment see [[display of a p-divisible group]].

### Examples

* The dual $\mu_{p^\infty}^D\simeq \mathbb{Q}_p/\mathbb{Z}_p$.

* For an abelian variety $X$, the dual is $X(p)^D=X^t(p)$ where $X^t$ denotes the dual abelian variety. Another proof that $X(p)$ has height $2g$ is to note that $X$ and $X^t$ have the same dimension $g$, so using our formula for height we get $ht(X(p))=2g$.

## Properties

The category of étale $p$-divisible groups is equivalent to the category of $p$-adic representations of the fundamental group of the base scheme .

## p-divisible groups and crystals

(...)

References: [Weinstein](#Weinstein)

## Relation to crystalline cohomology

(...)

## In derived algebraic geometry

See [Lurie](#pLurie). An important notion: [[nonstationary p-divisible group]]

## Related concepts

* Important tools in the study of $p$-divisible groups are [[Witt ring|Witt rings]], [[Dieudonné module|Dieudonné modules]] and more generally [[Dieudonné theory|Dieudonné theories]] assigning to a $p$-divisible group an object of [[linear algebra]] such as a [[display of a p-divisible group]].

* [[formal group]]

* [[group scheme]]

* [[Artin–Mazur formal group]]

* [[height of a variety]]

## Related entries

*  [[lectures on p-divisible groups|Lectures on p-divisible groups]]

## References

For references concerning [[Witt rings]] and [[Dieudonné modules]] see there.

### Original texts and classical surveys
* Barsotti, Iacopo (1962), "Analytical methods for abelian varieties in positive characteristic", Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain, pp. 77–85, MR 0155827

* Demazure, Michel (1972), [[lectures on p-divisible groups|Lectures on p-divisible groups]], Lecture Notes in Mathematics, 302, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060741, ISBN 978-3-540-06092-5, MR 034426, [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

* Grothendieck, Alexander (1971), "Groupes de Barsotti-Tate et cristaux", Actes du Congrès International des Mathématiciens (Nice, 1970), 1, Gauthier-Villars, pp. 431–436, MR 0578496

* Messing, William (1972), The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, 264, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058301, MR 0347836

* Serre, Jean-Pierre (1995) [1966], "Groupes p-divisibles (d'après J. Tate) [web](http://www.numdam.org/item?id=SB_1966-1968__10__73_0), Exp. 318", Séminaire Bourbaki, 10, Paris: Société Mathématique de France, pp. 73–86, MR 1610452

* Stephen Shatz, Group Schemes, Formal Groups, and $p$-Divisible Groups in the book Arithmetic Geometry Ed. Gary Cornell and Joseph Silverman, 1986

* Tate, John T. (1967), "p-divisible groups." [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.295.7566&rep=rep1&type=pdf), in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827 {#Tate}

### Modern surveys

* de Jong, A. J. (1998), Barsotti-Tate groups and crystals, "Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica II: 259–265, ISSN 1431-0635, MR 1648076

* Dolgachev, I.V. (2001), "P-divisible group" [web](http://www.encyclopediaofmath.org/index.php/P-divisible_group), in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

* Richard Pink, finite group schemes, 2004-2005, [pdf](http://www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf)

* Hoaran Wang, moduli spaces of p-divisible groups and period morphisms, Masters Thesis, 2009, [pdf](http://people.math.jussieu.fr/~dat/enseignement/M2HaoranWang.pdf)

* [[Jared Weinstein]], [[Weinstein, the geometry of Lubin-Tate spaces|the geometry of Lubi-Tate spaces]],  Lecture 1: Formal groups and formal modules, [pdf](http://www.math.ias.edu/~jaredw/FRGLecture.pdf){#Weinstein}

* Liang Xiao, notes on $p$-divisible groups, [pdf](http://math.uchicago.edu/~lxiao/files/notes/p-Divisble%20Groups.pdf)

### Further development of the theory

* Paul Goerss, p-divisible groups and Lurie's realization result, 2008, [pdf slides](http://www.math.ku.dk/~jg/homotopical2008/goerss.lec9.pdf)

* Jacob Lurie, [[A Survey of Elliptic Cohomology]], section 4.2, [pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf){#pLurie}

* Peter Scholze, Moduli of p-divisible groups, [arxiv](http://arxiv.org/abs/1211.6357)

* Thomas Zink, a dieudonné theory for p-divisible groups, [pdf](http://www.math.uni-bielefeld.de/~zink/CFTpaper.pdf)

* Thomas Zink, list of publications and preprints, [web](http://www.math.uni-bielefeld.de/~zink/z_publ.html)

* T. Zink, On the slope filtration, Duke Math. Journal, Vol.109 (2001), No.1, 79-95, [pdf](http://www.math.uni-bielefeld.de/~zink/slopes.pdf)

* T. Zink, the [[display of a p-divisible group|display of a formal p-divisible group]], to appear in [[Astérisque]], [pdf](http://www.math.uni-bielefeld.de/~zink/display.pdf)

* T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, [pdf](http://www.math.uni-bielefeld.de/~zink/Texel.pdf)

[[!redirects p-divisible group]]
[[!redirects p-divisible groups]]


p-local module

Let $G$ be a [[group]] and $M$ an [[abelian group]] with a $G$-action, i.e. a $\mathbb{Z}G$-[[module]] where $\mathbb{Z}G$ is the [[group ring]] of $G$.  Let $T$ be a set of [[prime numbers]]; we say that $M$ is a **$T$-local module** if for all $x\in G$, the element $(1 + x + x^2 + \cdots + x^{n-1})\in \mathbb{Z}G$ acts invertibly on $M$ for all $n$ not divisible by any primes in $T$.  If $T=\{p\}$ we speak of a $p$-local module.

Note that if $G$ acts trivially on $M$, this is equivalent to asking that $M$ is a $T$-local abelian group.  In general, it is strictly stronger.

The notion can also be characterized as a module over the [[localization of a ring|localization of the (noncommutative!) ring]] $\mathbb{Z}G$ at the set of elements $1 + x + x^2 + \cdots + x^{n-1}$.


p-norm

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### Context
#### Analysis
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[[!include analysis - contents]]
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# Contents
* table of contents
{: toc}

## Idea

For $p \in \mathbb{R}$, $p \geq 1$, the _$p$-norm_ is a [[norm]] on suitable [[real vector spaces]] given by the $p$th [[root]] of the [[sum]] (or [[integral]]) of the $p$th-[[powers]] of the [[absolute values]] of the vector components. With due care the definition makes sense for non-[[finite dimensional vector spaces]] such as [[sequence spaces]] and [[Lebesgue spaces]], making them into [[normed vector spaces]], hence [[metric spaces]].

For $p = 1$ the $p$-norm is the _Taxicab norm_ or _Manhattan norm_.

For $p = 2$ the $p$-norm is the standard _Euclidean norm_, defining [[Euclidean spaces]] and [[Hilbert spaces]] of [[square integrable functions]].

For $p = \infty$ the $p$-norm (found by taking the limit $p \to \infty$) is the [[supremum]] (or [[essential supremum]] in the continuous case) of the absolute values of the components of vectors, then called the _supremum norm_.

For $0 \leq p \lt 1$ one may still make sense of the formulas that define $p$-norms for $p \geq 1$ (see at _[Generalizations](#Generalizations)_ below), but the resulting concepts are no longer genuine norms.

## Definition

The concept of $p$-norm makes sense in increasing generality,

* _[For finite-dimensional vector spaces](#ThePNorm)_;

* _[For spaces of sequences of real numbers](#OnSequenceSpaces)_;

* _[For spaces of measurable functions](#OnLebesgueSpaces)_.

### The $p$-norm on finite dimensional vector spaces
 {#ThePNorm}

For $n \in \mathbb{N}$, $p \in \mathbb{R}$, $p \gt 0$, the _$p$-norm_ $\Vert - \Vert_p$ is the [[norm]] on the [[real vector space|real]] [[finite dimensional vector space]] $\mathbb{R}^n$ given by the $p$th [[root]] of the [[sum]] of the $p$-[[powers]] of the [[absolute value]] of the components of a given [[vector]] $\vec x = (x)_{i = 1}^n \in \mathbb{R}^n$:

$$
  {\Vert \vec x \Vert_p}  \coloneqq \root p {\sum_i {\vert x_i\vert^p}}
$$

Equipping it with this [[norm]] makes $\mathbb{R}^n$ a  _[[normed vector space]]_.

For $p = 2$ this is the _Euclidean norm_, the standard norm that defines [[Euclidean space]].

For $p = \infty$ one takes the [[supremum]] over the [[absolute values]] of the components

$$
  {\Vert \vec x\Vert_\infty}
    \;\coloneqq\;
  \underset{1 \leq i \leq n}{sup} {\vert x_i\vert}
  \,.
$$

The graphics on the right (from Wikipedia) shows unit circles in $\mathbb{R}^2$ with respect to various [[p-norms]].

### The $\mathcal{l}^p$-norm on sequence spaces
 {#OnSequenceSpaces}

For $p \in \mathbb{R}$, write $\ell^p$ for the vector space of those [[sequences]] $(x_i)_{i \in \mathbb{N}}$ in $\mathbb{R}$ for which the [[series]]

$$
  \underset{i \in \mathbb{N}}{\sum} {\vert x_i\vert}^p
    \;\lt\;
  \infty
$$

(the [[sum]] of the $p$th [[powers]] of the [[absolute value]] of the components of the sequence) [[convergence|converges]].

For $p \geq 1$ the the [[function]]

$$
  {\Vert -\Vert_p}
  \;\colon\;
  \ell^p
    \longrightarrow
  \mathbb{R}
$$

$$
  {\Vert (x_i)_{i \in \mathbb{N}} \Vert_p}
    \;\coloneqq\;
  \root{p}{\underset{i \in \mathbb{N}}{\sum} {\vert x_i\vert^p}}
$$

defines a [[norm]] on this [[real vector space]]. This [[normed vector space]] is [[complete metric space|complete]], hence a [[Banach space]]. This is called the _[[sequence space]]_.

For $p = \infty$ one takes $\ell^\infty$ to be the space of bounded sequences and

$$
  {\Vert (x_i)_{i \in \mathbb{N}} \Vert_\infty}
   \;\coloneqq\;
  \underset{k \in \mathbb{N}}{sup} {\vert x_k\vert }
$$

to be the [[supremum]] over the [[absolute values]] of the components of the sequence. This is also called the _supremum norm_.

### The $L^p$-norm on Lebesgue spaces
 {#OnLebesgueSpaces}

More generally, for $(X,\mu)$ a [[measure space]], write $L^p(X)$ for the [[vector space]] of [[equivalence classes]] of those [[measurable functions]] $f \colon X \to \mathbb{R}$, for which the [[integral]]

$$
  \int_X {\vert f \vert^p}  d\mu \;\lt\; \infty
$$

exists, and where two such functions are regarded as equivalent, $f_1 \sim f_2$, if

$$
  \int_X {\vert f_2 - f_1 \vert^p}  d\mu \;=\; 0
  \,.
$$

On this space the function

$$
  {\Vert -\Vert_p}
  \;\colon\;
  L^p(X) \longrightarrow \mathbb{R}
$$

$$
  {\Vert f \Vert_p}
  \;\coloneqq\;
  \root{p}{\int_X {\vert f\vert^p}} d\mu
$$

defines a [[norm]]. The [[triangle inequality]] holds due to [[Minkowski's inequality]].  The [[normed vector space]] $(L^p(X), {\Vert- \Vert_p})$ is also called a _[[Lebesgue space]]_.

## Generalizations for $0 \leq p \lt 1$
 {#Generalizations}

For $0 \leq p \lt 1$, the above definitions for $\Vert {-}\Vert_p$ still make sense in themselves, but the result is no longer a norm, as [[Minkowski's inequality]] (the [[triangle inequality]] for $p$-norms) fails.

A variant definition for $0 \lt p \leq 1$ (which agrees with the usual definition for $p = 1$, preserving continuity in $p$) leaves out the $p$th root; then the result satisfies the triangle inequality (and indeed is a [[metric]]) but fails to be a norm because it is not positive-homogeneous of degree $1$ (but of degree $p$ instead).  Such a thing is called an [[F-norm]].

For $p = 0$, we might try to take the limit as $p \searrow 0$.  For the unmodified $p$-norm (with the root), this is infinite if there is more than one nonzero entry and is the absolute value of the one nonzero entry if there is only one (or 0 if there is none); for the modified $p$-norm (without the root), it is the (possibly infinite) number of nonzero entries.  In either case, however the triangle inequality fails.  Therefore, there is a further modified $0$-norm, given by
$$ {\|(x_1, x_2, \ldots)\|_0} = \sum_{n=1}^\infty \frac {2^{-n} {|x_n|}} {1 + {|x_n|}} $$
for $l^0$, and this is an $F$-norm.  (But I don\'t know what is the justification for thinking of this as a $p$-norm for $p = 0$.)

## References

* Wikipedia, _[Lp space -- The p-norm in finite dimensions](https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions)_

* German Wikipedia, _[p-Norm](https://de.wikipedia.org/wiki/P-Norm)_

[[!redirects p-norm]]
[[!redirects p-norms]]

[[!redirects Lebesgue norm]]
[[!redirects Lebesgue norms]]

[[!redirects taxicab norm]]
[[!redirects taxicab norms]]

[[!redirects Manhattan norm]]
[[!redirects Manhattan norms]]

[[!redirects Euclidean norm]]
[[!redirects Euclidean norms]]

[[!redirects supremum norm]]
[[!redirects supremum norms]]

[[!redirects sup-norm]]
[[!redirects sup-norms]]


p-primary group

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## Definition

For $p$ a [[prime number]], a [[group]] is _$p$-primary_ if each of its elements $g$ has a [[prime power]] [[order of a group|order]] $p^{n(g)}$.

(Also called _primary group_ a _$p$-group_, but NO relation to _[[n-group]]_.)

## Properties

### Relation to finite abelian group

The [[fundamental theorem of finite abelian groups]] stats that every [[finite abelian group]] is a [[direct sum]] of its $p$-primary subgroups.

These are often called its _$p$-primary parts_ or _$p$-primary components_. See also at _[[Adams spectral sequence]]_ and for instance at _[[stable homotopy groups of spheres]]_.

### Nilpotency

+-- {: .num_prop #center}
###### Proposition
Every [[finite set|finite]] $p$-primary group $G$ has a nontrivial [[center of a group|center]] $Z(G)$.
=--

For a proof, see [[class equation]].

Since the center $Z(G)$ is a [[normal subgroup]] of $G$, we may define by induction (with the help of this proposition [here](/nlab/show/normal+subgroup#inverse)) a series of inclusions of normal subgroups $Z^k(G) \subseteq G$ where $Z^0(G)$ is the trivial subgroup and $Z^k(G)$ is the inverse image of the center $Z(G/Z^{k-1}(G))$ along the canonical homomorphism $G \to G/Z^{k-1}(G)$. The resulting series

$$Z^0(G) \subseteq \ldots \subseteq Z^{k-1}(G) \subseteq Z^k(G) \subseteq \ldots$$

is called the *upper central series* of $G$, and Proposition \ref{center} shows that in the case of a finite $p$-group, this series consists of strict inclusions that eventually terminate in the full subgroup $G$. A group with that property is a [[nilpotent group]]. In particular it is a [[solvable group]].

## Examples

* for the primary components of the [[stable homotopy groups of spheres]] see at _[homotopy groups of spheres -- Table](homotopy+groups+of+spheres#Tables)_.

## Related concepts

* [[stem]]

## References

* {#Sullivan} John Sullivan, _Classification of finite abelian groups_ ([pdf](http://www.isama.org/jms/m317/handouts/finabel.pdf))

[[!redirects p-primary groups]]

[[!redirects p-group]]
[[!redirects p-groups]]

[[!redirects primary group]]
[[!redirects primary groups]]

[[!redirects p-primary component]]
[[!redirects p-primary components]]


P-ring

Let $P$ be a [[commutative ring]] together with two types of structure on the set-valued functor $Hom(P, -)$ it [[representable functor|represents]]:

1.  a commutative ring structure, which means we can view it as an [[endofunctor]] on the category of commutative rings (so $Spec P$ is a commutative ring [[scheme]]), and
2.  a [[comonad]] structure on this endofunctor.

Then a **$P$-ring** is simply a commutative ring equipped with a [[co-action]] of this comonad.

This is similar to the situation for operators on abelian groups. Let $A$ be an [[abelian group]]; then $Hom(A, -)$ is already abelian-group-valued (since [[Ab]] is [[closed category|closed]]), so we don't need to do step (1). Then a comonad structure on the endofunctor $Hom(A, -)$ is the same as a (possibly non-commutative) ring structure on $A$.

The moral of the story is that operators on abelian groups naturally form rings, and operators on commutative rings naturally form $P$s. What you call such a $P$ is another matter. Borger and Wieland called such a thing a [[plethory]], with a wink at [[plethysm]], which is another name for composition of [[symmetric function]]s. Other people take a historical approach and use '[[Tall–Wraith monoid|Tall–Wraith]]' combined with some other words. Probably the most descriptive and generalizable name would be something like 'composition object in the category of commutative rings'.

Alternatively, the general naming scheme '[[plethystic object]] in the category $C$' could be used, where $C$ could be the category of commutative rings, the category of abelian groups, and so on.

##References##
*  J. Borger & B. Wieland, [Plethystic algebra](http://arxiv.org/abs/math/0407227)


p-torsion

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## p-torsion of abelian groups

Since any [[abelian group]] $G$ is a $\mathbb{Z}$-[[module]] we can form for any  $z\in \mathbb{N}$ the [[torsion subgroup]]

$$
  G[z]
  \,\coloneqq\,
  \big\{
    g
    \,\vert\,
    g \in G,
    \,
    z g = 0
  \big\}
  \,.
$$

Of particular interest in this article are those cases where $z = p^n$ for a [[prime number]] $p$ and a [[natural number]] $n$.

There are two important constructions to perform with these $G[p^n]$ namely taking [[limits]] and [[colimits]]:

$$
  S_p(G)
  \,\coloneqq\,
  colim_n G[p^n]
$$

$$
  T_p(G)
  \,\coloneqq\,
  lim_n G[p^n]
  \,.
$$

Here $S_p(G)$ is sometimes called the *$p$-[[torsion subgroup]]*; if $G$ is [[finite group|finite]] then $S_p(G)$ is also called *[[Sylow p-subgroup]] of $G$*.

$T_p(G)$ is called [[Tate module|p-adic Tate module]] of $G$.

Note that sometimes by "the Tate module" is meant a specific example of a Tate module. This example is mentioned [below](#PAdicTateModule).

## p-torsion of fields

$G[p]$ is obviously the [[kernel]] of the [[Frobenius]] [[endomorphism]] of $G$:

$$
  G[p]
  \,=\,
  \big(
    ker \, (g \mapsto g^p)
  \big)
  \,.
$$

In this form we can extend the Frobenius and hence this notion of $p$-torsion from [[abelian groups]] to [[fields]], if we require our field to be of [[characteristic]] $p$ such that we have $(a+b)^p = a^p + b^p$.

In fact the definition of $p$-torsion via the Frobenius has the advantage that we get additionally an adjoint notion to $p$-torsion which is sometimes called *Verschiebung*; this is explained at [[nLab:Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism
]].

## p-torsion of schemes

If $X$ denotes some [[scheme]] over a $k$-[[ring]] for $k$ being a [[field]] of [[characteristic]] $p$, we define its $p$-torsion component-wise by $X^{(p)}(R) \coloneqq X(s_* R)$.

## p-torsion of group schemes

Let $G$ is a commutative group scheme over a scheme $S$.  Define the multiplication by $p$ map as follows -   $[p] \colon G \xrightarrow{\Delta} G \times_S .... \times_S G \to G$.

  Because $G$ is a commutative group scheme, this is a map in the category of group schemes.  The $[p]$ torsion, $G[p]$, is then the pullback along the identity section of the multiplication by $[p]$ map.

The fiber of $G[p]$ at a given $s \in S$ is a group.  (Its a group scheme over the residue field of $s$).   And it is the $p$-torsion in the fiber of $G$ at $s$.

The notions of $S_p$, (as an Ind-scheme) and $T_p$ (as a scheme) readily generalize using this notion of $p$-torsion.

+-- {: .num_example #PAdicTateModule}
###### Example
(*the* $p$-adic [[Tate module]])

Let $G$ be a commutative group scheme over a field $k$ with separable closure $k^{sep}$.

Then $T_p\big(G(k^{sep})\big)$ is called *the $p$-adic Tate module of $G$*.
=--

This Tate module enters the [[Tate conjecture]].

If $G$ is an abelian variety $T_p(G(k^{sep}))$ is equivalently the first homology group of $G$.

## p-divisible groups

(main article: [[nLab:p-divisible group]])

Sometimes the information encoded in the colimit $T_p(G)=colim_n G[p^n]$ (we passed a contravariant functor from rings to schemes) is considered to be not sufficient and one wants more generally to study the codirected system

$$0\to G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots$$

itself. This system is called *$p$-divisible group of $G$*. Here $p$ denotes the multiplication-with-$p$ map.

We have

(1) The $G[p^i]$ are finite group schemes.

(2) The sequences of the form

$$0\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0$$

are exact.

(3) $G=\cup_j ker\, p^j\cdot id_G$

We have as cardinality (in group theory also called "rank") of the first item of the sequence $card \ker \,p=p^h$ for some natural number $h$. By pars pro toto we call $p^h$ also the rank of the whole sequence and $h$ we call its *height*.

Conversely we can define a $p$-divisible group to be a codirected diagram

$$G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}\dots$$

satisfying (1)(2)(3).

## Related concepts

* [[torsion approximation]]
* [[torsion points of an elliptic curve]]

## References

see the references at [[nLab:p-divisible group]], in particular the notes

* Richard Pink, *Shatz Group Schemes, Formal Groups, and $p$-Divisible Groups*

[[!redirects p-torsion subgroup]]
[[!redirects p-torsion subgroups]]


p-typical formal group law

## Properties

### Relation to Brown-Peterson spectrum

$p$-Typical formal groups are related to the [[Brown-Peterson spectrum]] as ordinary [[formal groups]] are related to [[MU]].

## Related concepts

* [[formal group law]]

* [[Brown-Peterson spectrum]]

## References

Section 19 of

* [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_, course notes ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf))

Appendix 2 of

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_

Code to compute the coefficients of $p$-typical formal groups

* [[Carl McTague]], [Computing _p_-Typical Formal Group Laws](https://www.mctague.org/carl/blog/2020/05/23/p-typical-fgl/)

[[!redirects p-typical formal group laws]]

[[!redirects p-typical formal group]]
[[!redirects p-typical formal group]]

[[!redirects p-typical formal group]]
[[!redirects p-typical formal groups]]