-adic cyclotomic character
Let be the absolute Galois group of . The purpose of this entry is to define, for every prime , a Galois representation:
Moreover, the restriction map is given by reduction modulo from to .
where the first map is simply restriction to and the second map is an isomorphism. By the remarks above, the representations are coherent in a strong sense, i.e.
Therefore, one can construct a “big” Galois representation:
by requiring , for every .
One can rephrase the above definition as follows. Let . We need to define a group homomorphism , so we need to first define and then check that it is a homomorphism. By the theory, is another primitive -th root of unity, thus
for some integer with (so is a unit modulo ). Moreover,
Therefore, modulo . Thus, we may define:
and as we have shown, is a unit of . Finally, the reader should check that is a group homomorphism.
|Title||-adic cyclotomic character|
|Date of creation||2013-03-22 15:36:16|
|Last modified on||2013-03-22 15:36:16|
|Last modified by||alozano (2414)|
|Synonym||-adic cyclotomic Galois representation|