fundamental character of level for the inertia group at
Let be the inertia subgroup of which we regard as a subgroup of via the injection above (for more information on the inertia subgroup at , , see the entry on Galois representations). Let be the finite field of elements. The purpose of this entry is to define -valued characters , for every :
which we will refer to as the fundamental character of level of .
Next, we define the fundamental characters in more generality. Let be the unique unramified field extension of degree (it is unique by local field theory). The residue field of is the field (because must be an extension of degree of ).
The field contains all th roots of unity.
Clearly, the polynomial has distinct roots in . Using Hensel’s lemma, one can check that each root in lifts to an element of . ∎
Notice that the fact that is unramified implies that the inertia group injects into . Therefore there is a map:
where the second map is simply given by restriction to .
The fundamental character of level is the map given by Eq. (1).
Note from the author: I would like to thank Eknath Ghate for explaining this to me.
|Title||fundamental character of level for the inertia group at|
|Date of creation||2013-03-22 15:36:26|
|Last modified on||2013-03-22 15:36:26|
|Last modified by||alozano (2414)|