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Compatible systems of mod \(p\) Galois representations and Hecke characters. (English) Zbl 1023.11026
Generalizing his results from [C. R. Acad. Sci., Paris, Sér. I 323, 117–120 (1996; Zbl 0868.11052)] the author shows that (for number fields \(K\) and \(L\)) an \(L\)-rational strictly compatible system of one-dimensional mod \(\wp\) representations \(\text{Gal}(\overline K / K)\rightarrow \text{GL}(1,{\mathcal O}_L/\wp)\) (where \(\wp\) runs through almost all finite places of \({\mathcal O}_L\)) necessarily comes from a Hecke character. The proof is purely algebraic and close to that in loc.cit., avoiding a slight inaccuracy of this. Furthermore, class field theory is used to get rid of an extra condition from loc.cit. (boundedness of the conductor). The author uses his one-dimensional result in order to motivate the conjecture that strictly compatible \(L\)-integral systems of mod \(\wp\) representations are “motivic”; he makes this notion precise in a more detailed conjecture.

11F80 Galois representations
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