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On residually reducible representations on local rings. (English) Zbl 0996.16007

Let \(A\) be a local Artinian ring with maximal ideal \(\mathcal M\), let \(R\) be an \(A\)-algebra and let \(\rho\) be an \(A\)-representation of \(R\). If the residual representation \(\overline\rho\) (that is, with values in the residue field) is absolutely irreducible, then it is well-known by a result of Carayol that \(\rho\) is completely determined by its trace. This is no longer true in the reducible case. The author proves a version of this result assuming some further hypothesis.

MSC:

16G10 Representations of associative Artinian rings
16P20 Artinian rings and modules (associative rings and algebras)
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References:

[1] Carayol, H., Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, Contemp. Math., 165, 213-235 (1994) · Zbl 0812.11036
[2] E. Urban, Selmer groups and the Eisenstein-Klingen ideal, Oct. 1997; E. Urban, Selmer groups and the Eisenstein-Klingen ideal, Oct. 1997 · Zbl 1061.11027
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