One reason we prefer the Weil group over the Galois group (at least in the local case) is that the Weil group is locally compact, thus it has "more" representations (over $\bf C$). In fact, all $\bf C$-valued characters of $Gal(\bar{\bf Q_p} / \bf Q_p)$ have finite image, where as that of $W_{\bf Q_p}$ can very well have infinite image. The same goes for general representations of these groups (recall that $\bf{GL}_n(\bf C)$ has no small subgroups.)
The global Weil group (which is much more complicated than the local one), on the other hand, is a rather mysterious object that is pretty much untouched in modern number theory as far as I can tell. Supposedly the global Langlands group used in the global Langlands correspondence should be the extension of the global Weil group by a compact group, but this is still largely conjectural.
The standard reference is Tate's "Number Theory Background" in the Corvallis volumes (available for free at ams.org). Also Brooks Roberts has notes on Weil representations available at his website.
Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening.
Since I had read and enjoyed Lazard’s paper on one-dimensional formal group (laws), which dealt with the case of a base field of characteristic $p$, I decided to look at formal groups over $p$-adic rings. For whatever reason, I wanted to know about the endomorphism rings of these things, and gradually recognized the similarity between, on the one hand, the case of elliptic curves and their supersingular reduction mod $p$, when that phenomenon did occur, and, on the other hand, formal groups over $p$-adic integer rings of higher height than $1$.
I had taken, or sat in on, Tate’s first course on Arithmetic on Elliptic Curves, and was primed for all of this. In addition, I was aware of Weierstrass Preparation, and the power it gave to anyone who wielded it. And in the attempt to prove a certain result for my thesis, I had thought of looking at the torsion points on a formal group, and I suppose it was clear to me that they formed a module over the endomorphism ring. Please note that it was not my idea at all to use them as a representation module for the Galois group.
But Tate was looking over my shoulder at all times, and no doubt he saw all sorts of things that I was not considering. At the time of submission of my thesis, I did not have a construction of formal groups of height $h$ with endomorphism ring $\mathfrak o$ equal to the integers of a local field $k$ of degree $h$ over $\Bbb Q_p$. Only for the unramified case, and I used extremely tiresome degree-by-degree methods based on the techniques of Lazard. Some while after my thesis, I was on a bus from Brunswick to Boston, and found not only that I could construct formal groups in all cases that had this maximal endomorphism structure, but that one of them could take the polynomial form $\pi x+x^q$. Tate told me that when he saw this, Everything Fell Into Place. The result was the wonderful and beautiful first Lemma in our paper, for which I can claim absolutely no responsibility. My recollection, always undependable, is that the rest of the paper came together fairly rapidly. Remember that Tate was already a master of all aspects of Class Field Theory. But if the endomorphism ring of your formal group is $\mathfrak o$ and the Tate module of the formal group is a rank-one module over this endomorphism ring, can the isomorphism between the Galois group of $k(F[p^\infty]])$ over $k$ and the subgroup $\mathfrak o^*\subset k^*$ fail to make you think of the reciprocity map?
Best Answer
The reciprocity map is completely natural (in the technical sense of category theory). For example, if $K$ and $L$ are two local fields, and $\sigma:K \rightarrow L$ is an isomorphism, then $\sigma$ induces an isomorphism of multiplicative groups $K^{\times} \rightarrow L^{\times}$ and also of abelian absolute Galois groups $G\_K^{ab} \rightarrow G\_L^{ab}$. The reciprocity laws for $K$ and $L$ are then compatible with these two isomorphisms induced by $\sigma$.
On the other hand the factorization $K^{\times} = {\mathbb Z} \times {\mathcal O}\_K^{\times}$ is not canonical (it depends on a choice of uniformizer), and the identification of ${\mathcal O}\_K^{\times}$ with the Galois group of a maximal totally ramified abelian extension of $K$ also depends on a choice of uniformizer (which goes into the construction of the Lubin--Tate formal group, and hence into the construction of the totally ramified extension; different choices of uniformizer will give different formal groups, and different extensions).
As others pointed out, the local reciprocity map is also a logical consequence of the global Artin map and global Artin reciprocity law (which makes no reference to local multiplicative groups, but simply to the association $\mathfrak p \mapsto Frob\_{\mathfrak p}$ of Frobenius elements to unramified prime ideals; see the beginning of Tate's article in Cassels--Frohlich for a nice explanation of this). Thus it is natural in a more colloquial sense of the word as well.
Indeed, the idelic formulation of the glocal reciprocity map and the formulation of the local reciprocity map in terms of multiplicative groups are not accidental or ad hoc inventions; they were forced on number theorists as a result of making deep investigations into the nature of global class field theory.