On modular Galois representations modulo prime powers
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On modular Galois representations modulo prime powers. / Chen, Imin; Kiming, Ian; Wiese, Gabor.
In: International Journal of Number Theory, Vol. 9, No. 1, 2013, p. 91-113.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - On modular Galois representations modulo prime powers
AU - Chen, Imin
AU - Kiming, Ian
AU - Wiese, Gabor
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
U2 - 10.1142/S1793042112501254
DO - 10.1142/S1793042112501254
M3 - Journal article
VL - 9
SP - 91
EP - 113
JO - International Journal of Number Theory
JF - International Journal of Number Theory
SN - 1793-0421
IS - 1
ER -
ID: 44022178