### Context

#### Algebra

higher algebra

universal algebra

# $p$-adic numbers

## Idea

For $p$ any prime number, the $p$-adic numbers (or $p$-adic rational numbers, for emphasis) form a field $\mathbb{Q}_p$ that 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).

There have long been speculations (see the references below) 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

and then consider

### Recollection of the $p$-adic integers

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 $p^m \mathbf{Z}\subset p^n\mathbf{Z}$ which induces the canonical homomorphism of quotients $\phi_{n,m}:\mathbf{Z}/p^n\mathbf{Z}\to \mathbf{Z}/p^m\mathbf{Z}$. These homomorphism 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 (Stone) space and it is in particular a compact 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 threads, 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 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 dual to $\mathbf{Z}(p^\infty)$. Observe that $\mathbf{Z}(p^\infty)$ embeds in $S^1$ as the set of all 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

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

###### 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 completions of the rational numbers: the real numbers and the $p$-adic numbers.

###### 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

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) 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 dual to the discrete 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

function field analogy

number fields (“function fields of curves over F1”)function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
$\mathbb{Z}$ (integers)$\mathbb{F}_q[z]$ (polynomials, function algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
$\mathbb{Q}$ (rational numbers)$\mathbb{F}_q(z)$ (rational functions)meromorphic functions on complex plane
$p$ (prime number/non-archimedean place)$x \in \mathbb{F}_p$$x \in \mathbb{C}$
$\infty$ (place at infinity)$\infty$
$Spec(\mathbb{Z})$ (Spec(Z))$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)complex plane
$Spec(\mathbb{Z}) \cup place_{\infty}$$\mathbb{P}_{\mathbb{F}_q}$ (projective line)Riemann sphere
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)$\frac{\partial}{\partial z}$ (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
$\mathbb{Z}_p$ (p-adic integers)$\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$)$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)formal disks in $X$
$\mathbb{Q}_p$ (p-adic numbers)$\mathbb{F}_q((z-x))$ (Laurent series around $x$)$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
$\mathcal{O}_K$ (ring of integers)$\mathcal{O}_{\Sigma}$ (structure sheaf)
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)$\Sigma$ (arithmetic curve)$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)$\frac{\partial}{\partial z}$
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
$v$ prime ideal in ring of integers $\mathcal{O}_K$$x \in \Sigma$$x \in \Sigma$
$K_v$ (formal completion at $v$)$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
$\mathcal{O}_{K_v}$ (ring of integers of formal completion)$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
$\mathbb{A}_K$ (ring of adeles)$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
$\mathcal{O}$$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
Galois group$\pi_1(\Sigma)$ fundamental group
Galois representationflat connection (“local system”) on $\Sigma$
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functionsHasse-Weil zeta function

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

Review in the context of p-local homotopy theory is in

• 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)

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

• David Lubicz, An introduction to the algorithmic of $p$-adic numbers (pdf)

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

$p$-adic differential equations are discussed in

The development of rigid analytic geometry starts with

• John Tate, Rigid analytic spaces, Invent. Math. 12:257–289, 1971.

p-adic homotopy theory is discussed in