Why does taking completions make number fields simpler?

Question

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example,

• If $K$ is a finite extension of $\mathbb Q_p$, then $\mathcal O_K$ is a DVR, whereas in the number fields case, $\mathcal O_K$ is just a Dedekind Domain, and need not even be a UFD.

• There are only finitely many extensions of given degree of a finite extension of $\mathbb Q_p$

• If $L/K$ is are extensions of $\mathbb Q_p$, both with normalised absolute values, and $m$ is a prime ideal of $\mathcal O_K$ (i.e. the unique maximal ideal), then there is exactly one prime ideal lying over $m$. Unlike in the number fields case, where a prime ideal $\mathcal P$ can have multiple distinct primes lying over it.

My question is

What is it about taking completions that makes the picture so much simpler?

I would be interested in seeing examples where the fact that $\mathbb Q$ is not complete causes extra complications such as those mentioned above.

Answer 1

I can add further alebro-geometric intuition, if you'd like (as in the above comments), but here's one concrete difference:

Theorem: Let $F$ be a complete field, and $V$ a finite dimensional $F$-space. Then, all norms on $F$ are equivalent.

Now, let $L/F$ be an extension. Then, inequivalent norms on $L$ correspond to different primes over $\mathfrak{p}$ (here $\mathfrak{p}$ is the prime defining the absolute value on $F$), this topological fact forces there to only be one prime over $\mathfrak{p}$!

Answer 2

Allow me to add a few comments on the questions/answers (including the ones here) around the « local-global principle » and which concern, depending on taste and interest, many deep phenomena and subterranean connections in number theory. Since this is a quick survey, I hope you’ll forgive me some imprecisions or inacurracies.

« Why does taking completion make number fields simpler ? ». One first - only skin deep - answer would be : « Because the ring of integers of the local (p-adic) field becomes principal », which allows to get rid right away of one of the main technical problems in the arithmetic of global number rings (example : if the ring of integers of the p-th cyclotomic field were always principal, Fermat’s LT would be easy to prove). From that point of view, one could however argue that taking completion is somewhat overkilling, since it eliminates one major arithmetical object, which is the class group. Not quite so : taking completion at infinite primes allows to do complex analysis and introduce what has been nicknamed « the DNA » of number fields, namely the complex L-functions, and the class group makes its come back under the guise of the classical Dedekind analytic class number formula. Taking completion at finite primes allows also to : 1. Do p-adic analysis and construct p-adic L-functions which interpolate (see below) the complex ones in a certain way. 2. Give a precise relationship between the p-adic L-function and the « characteristic series » of the projective limit of the p-class groups along the tower obtained by adding to the base field all the $p^n$-th roots of unity. This is the so called Main Conjecture of Iwasawa theory, now a theorem of Mazur-Wiles and of Wiles.

The MC above being far from « simple », a modified question would be : « What kind of global (arithmetic) information can be obtained by taking completions ? ». Of course not all problems in number theory could be solved in such a way, but quite a few can, and they are not skin deep . The local-global approach here should be understood in the sense of differential geometry, where the description of a global variety is given by its atlas, the collection of all its local charts. It implies in particular to study simultaneously all the local problems, at all finite as well as infinite primes. This is already visible in the formula product. But among an impressive list (see e.g. the « List of local to global principles » asked for by Quanta), perhaps the most elaborate example concerns class field theory. Nowadays CFT is taught by introducing first local CFT, then derive global CFT from the local one(s) using the technique of idèles. In his « History of CFT », H. Hasse stated that « one can say that by Chevalley’s ideas (not to say : idèles), the local-global principle has taken place in class field theory ». Coming back to the local part, recall that all the primes, finite or infinite, must be treated equally and simultaneously . For instance, although local CFT deals essentially with non archimedean local fields, the archimedean theory is also present, even if it is trivial. About the role of infinite primes in number theory and Zhen Lin’s question « In what senses are archimedean places infinite ? », it has been rightly noted that the most precise modern form of the analogy between algebraic fields and algebraic curves over finite fields lies in Arakelov’s theory. See also the speculative question raised by Pavel Coupek in « Infinite primes (places) of a number field geometrically », about a construction that would make naturally the set of all (finite and infinite) primes of a number field F into a scheme ; namely, so that the set of all finite primes would form an an open subscheme isomorphic to Spec $O_F$, the infinite primes « completing » somehow the curve Spec $O_F$ . But something deeper than just an analogy (however elaborate) is required to explain the paradox contained e.g. in the class number formula : on the one side we have a group and a discriminant, which are discrete objects (thinking of it, as are all the objects dealt with by algebraic number theory) ; on the other we have a transcendental regulator and the special value of an analytic function. The Iwasawa MC (Wiles’ theorem) appears equally paradoxical , because it blends the complex word (which is connected) together with the p-adic world (which is totally disconnected). Actually a conceptual explanation is already available in Grothendieck’s theory (as yet incomplete) of « motives ». A motive is supposed to be a certain piece in the cohomology of a smooth projective variety, to which can be attached a « motivic cohomology » which is universal in the sense that it gives rise to the usual cohomologies of Betti, de Rham and p-adic étale, called « realizations ». There are comparison theorems which say (roughly speaking) that these become isomorphic after tensorizing with C or Q_p, but when doing number theory or arithmetic geometry, we must stay at the level of a number field. « Pure motives » have been suitably defined by Grothendieck, but the general, so called « mixed motives » are still in limbo. However Voevodsky has succeeded, starting from the notion of homotopy of algebraic varieties, in defining their motivic cohomology, which is sufficient for the purpose of explaining the arithmetic meaning of the special values of L-functions (recall that these are the DNA of number fields). In order not to go too far, let us stick to the special values of the Dekind zêta function attached to a number field F. The classical theorem of Siegel-Klingen asserts that the values or $zêta_F$ at the negative integers (called « special values ») are rational, and can be interpolated (this is a theorem due to many people) by a meromorphic p-adic function, $zêta_{p, F}$. Thus, because Q is contained in both C and Q_p, these « special values » are the meeting points of the p-adic world and the archimedean world. It remains to explain precisely how these interact.The motive which comes into play is the « Tate motive » M = Q(m) (this is just a notation), m ∈ Z ; for m positive, its motivic cohomology happens to coincide in degree 1 with the Quillen K-groups $K_{2m-1}$(F) (whose definition comes from algebraic topology) tensored with Q. The Galois representation afforded by the motivic cohomology allows to construct the motivic L-function L(M,s) following a recipe of Serre, Deligne, etc. To understand the special values of L(M,s) at s = 1 – m, m positive, one goes to the Betti and étale realizations via « regulator maps » : the archimedean regulator of Borel-Beilinson is a far reaching generalization of the Dedekind regulator, and the Soulé étale regulator is constructed using p-adic Chern classes. Moreover, if F/Q is abelian, the group $K_{2m-1}$(F) contains a special element, the « Beilinson element », which goes under the Beilinson regulator (resp. the étale regulator) to a combination of special values of polylog functions – this is where complex analysis comes into play (resp. to the « Deligne-Soulé element », constructed from cyclotomic units – this is where Iwasawa theory comes into play). The simple fact that the Beilinson element is the common source yields the generalization to all negative arguments 1 – m of the class number formula (modulo some calculations of course). Naturally, all this process is expected to hold for a non abelian F. Actually the whole stuff can be extended to the special values of general motivic L-functions : this is nowadays called the « Tamagawa number conjecture », due to Bloch and Kato.
To summarize : it is possible to skip the technical details above and still explain « philosophically » the (arithmetic) interplay between finite and infinite primes. Just adapt the Platonician apologue of the cave : the multiple cohomologies floating around algebraic varieties are but shadows cast on the walls of the cave by the light of the sun behind our back. We are vaguely aware of analogies between the shadows (e.g. the comparison theorems), but by nature there cannot be « horizontal » links between them. The genuine relations are « vertical », and we cannot understand them unless we return to the object which projects its shadows, and which Plato calls the « archetype » (here, motivic cohomology, the regulators being the rays of the sun) . ¤