On modular Galois representations modulo prime powers

Imin Chen; Ian Kiming; Gabor Wiese

doi:10.1142/S1793042112501254

On modular Galois representations modulo prime powers

Imin Chen, Ian Kiming, Gabor Wiese

Institut for Matematiske Fag

7 Citationer (Scopus)

1098 Downloads (Pure)

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.

Originalsprog	Engelsk
Tidsskrift	International Journal of Number Theory
Vol/bind	9
Udgave nummer	1
Sider (fra-til)	91-113
ISSN	1793-0421
DOI	https://doi.org/10.1142/S1793042112501254
Status	Udgivet - feb. 2013

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10.1142/S1793042112501254

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Citationsformater

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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.",

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