I'm trying to explicitly compute modular representations of some finite groups -- the easiest example to discuss is the cyclic group $C_3$ when $p=3$. The three ordinary irreducible modules for $C_3$, which can be realised over the Eisenstein rationals $\mathbb{Q}(\omega)$, are all one-dimensional and have characters $1,\omega$ and $\omega^2$ respectively.
I want to know how to "reduce" these three representations modulo 3. I know from general theory that there is only one simple module for $C_3$ in characteristic three (the trivial representation), and I also know that each of the three irreducibles above should reduce to the trivial representation.
The general theory of Brauer etc says to choose a field $K$ which is complete with respect to some valuation, whose valuation ring $\mathcal{O}$ has residue field $k$ with characteristic 3. We then perform a reduction procedure to turn representations over $K$ into representations over $k$.
I understand we need to take $K=\mathbb{Q}(\omega)$.
- Do we need to take our valuation $\nu$ to be the 3-adic valuation?
- In that case, what is the 3-adic valuation of an arbitrary element of $K$?
- What is $\mathcal{O}$? (This should be the set of elements $x$ with $\nu(x)\geq 0$?)
- What is the maximal ideal $\mathfrak{m}$ of $\mathcal{O}$? (This should be the set of elements $x$ with $\nu(x)>0$?)
- What is $k$?
- How do we perform this reduction procedure to show that the three representations above all reduce to the trivial representation?