symmetric monoidal (∞,1)-category of spectra
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
For $p$ any prime number, the $p$-adic numbers $\mathbb{Q}_p$ (or $p$-adic rational numbers, for emphasis) are a field that completes the field of rational numbers. 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.
$p$-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.
We first recall the definition and construction of the p-adic integers
and then consider
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 (inside the category 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 $\ldots a_n a_{n-1} \ldots a_1 a_0$.
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 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.
An element $x\in \mathbb{Z}_p$ is invertible precisely if $x_0 \neq 0$.
Ostrowski's theorem implies there are precisely two kinds of completions of the rational numbers: the real numbers and the $p$-adic numbers.
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.
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.
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.
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 = 0 | genus 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 function | Goss 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 field | genus of an algebraic curve | genus 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 representation | “ | flat connection (“local system”) on $\Sigma$ | |
class field theory | |||
class field theory | “ | geometric class field theory | |
Hilbert reciprocity law | Artin reciprocity law | Weil 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 correspondence | function field Langlands correspondence | geometric 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 fields | Tamagawa-Weil for function fields | ||
theta functions | |||
Hecke theta function | functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$ | ||
zeta functions | |||
Dedekind zeta function | Weil zeta function | zeta function of a Riemann surface/of the Laplace operator on $\Sigma$ | |
higher dimensional spaces | |||
zeta functions | Hasse-Weil zeta function |
natural number, integer, rational number, algebraic number, Gaussian number, irrational number, real number
The $p$-adic numbers had been introduced in
A standard reference is
Review in the context of p-local homotopy theory is in
Review of the use of $p$-adic numbers in arithmetic geometry includes
A formalization in homotopy type theory and there in Coq is discussed in
$p$-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
p-adic homotopy theory is discussed in