$p$-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

• Recollection of the p-adic integers

and then consider

• The p-adic numbers proper

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 $p^n \mathbf{Z}\subset p^m\mathbf{Z}$ 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 (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

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

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

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

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

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

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.

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

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

fits into an isomorphism of exact sequences

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

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

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-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

• Álvaro Pelayo, Vladimir Voevodsky, Michael Warren, A preliminary univalent formalization of the p-adic numbers (arXiv:1302.1207)

$p$-adic differential equations are discussed in

• Kiran Kedlaya, $p$-adic differential equations (pdf, course notes)

• Gilles Cristol, Exposants $p$-adiques et solutions dans les couronnes (pdf)

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

• Jacob Lurie, Rational and p-adic Homotopy Theory