Section 4.5 Making the reciprocity map explicit
It is natural to ask whether the local reciprocity map can be described more explicitly. In fact, given an explicit cocycle generating we can trace through the arguments to get the local reciprocity map. However, the argument is somewhat messy, so I won’t torture you with all of the details; the point is simply to observe that everything we’ve done can be used for explicit computations. (This observation is apparently due to Dwork.)
If you find this indigestible, you may hold out until we hit abstract class field theory. That point of view will give a different (though of course related) mechanism for computing the reciprocity map (see Section 5.2).
Subsection Initial setup
and apply the “snaking” construction: pull back to take the norm to get (switching to multiplicative notation). The and term cancel out when you take the product, so we get as the inverse image of
As noted above, one needs to make this truly explicit; one can get using explicit generators of if you have them. For one can use roots of unity; for general one can use the Lubin-Tate construction. Alternatively, one can argue as in our proof that is cyclic of order see below.
Subsection An explicit cocycle via periodicity
Now is isomorphic to which is generated by a uniformizer To explicate that isomorphism, we recall generally how to construct the isomorphism for cyclic with a distinguished generator Recall the exact sequence we used to produce the isomorphism in Theorem 3.4.1:
(Remember, acts on both factors in The first map is the second is and the third is ) Let be the kernel of the third arrow, so and are exact.
Subsection From a cocycle to reciprocity
Back to the desired computation. Applying this to acting on with the canonical generator equal to the Frobenius, we get that is generated by a cocycle with if and 1 otherwise. Now push this into the general theory says the image comes from That is, for let be the integer such that restricted to equals Then there exists a 1-cochain such that belongs to and depends only on the images of in Putting we thus have
The upshot: once you compute such a (which I won’t describe how to do, since it requires an explicit description of ), to find the inverse image of under the Artin map, choose a lift of into then compute