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On modular Galois representations modulo prime powers

Imin Chen, Ian Kiming, Gabor Wiese

Number 3
Author Gabor Wiese
Year 2011

We study modular Galois representations mod pm. We show that there are three progressively weaker notions of modularity for a Galois representation mod pm: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M.
Using results of Hida we display a `stripping-of-powers of p away from the level' type of result: A mod pm strongly modular representation of some level Npr is always dc-weakly modular of level N (here, N is a natural number not divisible by p).
We also study eigenforms mod pm corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod pm to any `dc-weak' eigenform, and hence to any eigenform mod pm in any of the three senses.
We show that the three notions of modularity coincide when m=1 (as well as in other, particular cases), but not in general.

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