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

Harvard

Chen, I, Kiming, I & Wiese, G 2013, 'On modular Galois representations modulo prime powers', International Journal of Number Theory, vol. 9, no. 1, pp. 91-113. https://doi.org/10.1142/S1793042112501254

APA

Chen, I., Kiming, I., & Wiese, G. (2013). On modular Galois representations modulo prime powers. International Journal of Number Theory, 9(1), 91-113. https://doi.org/10.1142/S1793042112501254

Vancouver

Chen I, Kiming I, Wiese G. On modular Galois representations modulo prime powers. International Journal of Number Theory. 2013;9(1):91-113. https://doi.org/10.1142/S1793042112501254

Author

Chen, Imin ; Kiming, Ian ; Wiese, Gabor. / On modular Galois representations modulo prime powers. In: International Journal of Number Theory. 2013 ; Vol. 9, No. 1. pp. 91-113.

Bibtex

@article{0abb61bfb75348b8a20f240dc3688213,
title = "On modular Galois representations modulo prime powers",
abstract = "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.",
author = "Imin Chen and Ian Kiming and Gabor Wiese",
year = "2013",
doi = "10.1142/S1793042112501254",
language = "English",
volume = "9",
pages = "91--113",
journal = "International Journal of Number Theory",
issn = "1793-0421",
publisher = "World Scientific Publishing Co. Pte. Ltd.",
number = "1",

}

RIS

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