symmetric monoidal (∞,1)-category of spectra
For any prime number, the -adic numbers (or -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 -adic numbers form a non-archimedean field.
We first recall the definition and construction of the p-adic integers
and then consider
Let now be a prime number. Then for any two positive integers there is an inclusion which induces the canonical homomorphism of quotients . These homomorphism for all pairs 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 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 has a natural topology as a profinite (Stone) space and it is in particular a compact Hausdorff topological ring. More concretely, is the closed (hence compact) subspace of the cartesian product of discrete topological spaces (which is by the Tihonov theorem compact Hausdorff) consisting of threads, i.e. sequences of the form with and satisfying .
The kernel of the projection , to the -th component (which is the corresponding projection of the limiting cone) is , i.e. the sequence
An element in is invertible (and called a -adic unit) iff is not divisible by .
Let be the group of all invertible elements in . Then every element can be uniquely written as with and . The correspondence defines a discrete valuation called the p-adic valuation and is said to be the -adic valuation of . Of course, as required by the axioms of valuation. The norm induced by the valuation is (up to equivalence) given by , and this in turn induces a metric
Concretely, a -adic integer may be written as a base- expansion
with . Addition and multiplication are performed with carrying as in ordinary base- arithmetic, carried infinitely far to the left if is written as .
Algebraically, the ring of -adic integers is isomorphic to the endomorphism ring where is the Prüfer group . In particular, is tautologically a -module.
The field of -adic numbers is the field of fractions of the p-adic integers . The -adic valuation extends to a discrete valuation, also denoted on . Indeed, it is still true for all that they can be uniquely written in the form where (the same group as before), but now one needs to allow . One defines the metric on by the same formula as for . It appears that is a complete field (in particular locally compact Hausdorff) and that is an open subring.
The distance satisfies the “ultrametric” inequality
Concretely, a -adic number may be written as , with only finitely many negative powers of occurring. If , 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 is invertible precisely if .
For that reason the naive idea of formulating p-adic geometry in analogy to complex analytic geometry as modeled on domains in , 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 -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 is dual to the discrete Prüfer group that is isomorphic to a direct limit of finite cyclic groups . The canonical inclusion induces an isomorphism , in fact an isomorphism of -modules, so there is an exact sequence
This exact sequence is Pontrjagin self-dual in the sense that the map induced from the pairing
fits into an isomorphism of exact sequences
where the commutativity of the squares can be traced to the fact that is a -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 (arithmetic curves)||Riemann surfaces/complex curves|
|affine and projective line|
|(integers)||(polynomials, function algebra on affine line )||(holomorphic functions on complex plane)|
|(rational numbers)||(rational functions)||meromorphic functions on complex plane|
|(prime number/non-archimedean place)|
|(place at infinity)|
|(Spec(Z))||(affine line)||complex plane|
|(projective line)||Riemann sphere|
|(Fermat quotient)||(coordinate derivation)||“|
|genus of the rational numbers = 0||genus of the Riemann sphere = 0|
|(p-adic integers)||(power series around )||(holomorphic functions on formal disk around )|
|(“-arithmetic jet space” of at )||formal disks in|
|(p-adic numbers)||(Laurent series around )||(holomorphic functions on punctured formal disk around )|
|(ring of adeles)||( adeles of function field )||(restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)|
|(group of ideles)||( ideles of function field )|
|Jacobi theta function|
|Riemann zeta function||Goss zeta function|
|branched covering curves|
|a number field ( a possibly ramified finite dimensional field extension)||a function field of an algebraic curve over||(sheaf of rational functions on complex curve )|
|(ring of integers)||(structure sheaf)|
|(spectrum with archimedean places)||(arithmetic curve)||(complex curve being branched cover of Riemann sphere)|
|(lift of Frobenius morphism/Lambda-ring structure)||“|
|genus of a number field||genus of an algebraic curve||genus of a surface|
|prime ideal in ring of integers|
|(formal completion at )||(function algebra on punctured formal disk around )|
|(ring of integers of formal completion)||(function algebra on formal disk around )|
|(ring of adeles)||(restricted product of function rings on all punctured formal disks around all points in )|
|(function ring on all formal disks around all points in )|
|(group of ideles)|
|Galois group||“||fundamental group|
|Galois representation||“||flat connection (“local system”) on|
|class field theory|
|class field theory||“||geometric class field theory|
|Hilbert reciprocity law||Artin reciprocity law||Weil reciprocity law|
|(idele class group)||“|
|“||(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|
|(constant sheaves on this stack form unramified automorphic representations)||“||(moduli stack of bundles on the curve , by Weil uniformization theorem)|
|Tamagawa-Weil for number fields||Tamagawa-Weil for function fields|
|Hecke theta function||functional determinant line bundle of Dirac operator/chiral Laplace operator on|
|Dedekind zeta function||Weil zeta function||zeta function of a Riemann surface/of the Laplace operator on|
|higher dimensional spaces|
|zeta functions||Hasse-Weil zeta function|
The -adic numbers had been introduced in
A standard reference is
Review of the use of -adic numbers in arithmetic geometry includes
-adic differential equations are discussed in
The development of rigid analytic geometry starts with
p-adic homotopy theory is discussed in