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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Ian Kiming (Copenhagen, DNK)

On modular Galois representations modulo prime powers

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.

This is joint work with I. Chen (Simon Fraser) and G. Wiese (Duisburg-Essen).