On modular Galois representations modulo prime powers
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- ON MODULAR GALOIS REPRESENTATIONS
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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 level-lowering result ("stripping-of-powers of p away from the level"): 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.
Originalsprog | Engelsk |
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Tidsskrift | International Journal of Number Theory |
Vol/bind | 9 |
Udgave nummer | 1 |
Sider (fra-til) | 91-113 |
ISSN | 1793-0421 |
DOI | |
Status | Udgivet - 2013 |
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