# 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 p^{m}. We show that there are
three progressively weaker notions of modularity for a Galois representation
mod p^{m}: 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 p^{m }strongly modular representation of some
level Np^{r} is always dc-weakly modular of level N (here, N is a natural
number not divisible by p).

We also study eigenforms mod p^{m} 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 p^{m} to any `dc-weak' eigenform,
and hence to any eigenform mod p^{m} 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.