Examples

Example: generic (ex-b604c9)

We create two \(\varphi\)-modules \(D_1, D_2\), build their direct sum, and compute its slopes and corresponding representation.

> k<e9> := GF(3,2);
> S<u> := LaurentSeriesRing(k,20);
> D1 := ElementaryPhiModule(S,3,2);
> D1;
Phi-module of dimension 3 over Laurent series field in u over GF(3^2)
with fixed relative precision 20 with matrix
[ O(u^20)         O(u^20)        u^2 + O(u^20)]
[ 1 + O(u^20)     O(u^20)             O(u^20) ]
[ O(u^20)        1 + O(u^20)          O(u^20) ] and Frobenius [1,3]
> M := Matrix(S,2,2,[0,k.1*u,1,0]);
> D2 := PhiModule(M);
> D2;
Phi-module of dimension 2 over Laurent series field in u over GF(3^2)
with fixed relative precision 20 with matrix
[ O(u^20)       e9*u^2 + O(u^20)]
[ 1 + O(u^20)            O(u^20) ] and Frobenius [1,3]
> D := DirectSum(D1,D2);
> Slopes(D);
[
  [2, 1],
  [3, 2]
]
> SSGaloisRepresentation(D);
Semisimple representation of the absolute Galois group of
Laurent series field in u over GF(3^2) with fixed relative
precision 20 with coefficients in Finite field of size 3 and
components [
[3, 18],
[2, 3]
]

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