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Elliptic Curves, Modular Forms and Iwasawa Theory - Conference in honour of the 70th birthday of John Coates

JHC70 2015: Elliptic Curves, Modular Forms and Iwasawa Theory pp 469–492Cite as

Bigness in Compatible Systems

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 188))

Abstract

Clozel, Harris and Taylor have recently proved a modularity lifting theorem of the following general form: if \(\rho \) is an \(\ell \)-adic representation of the absolute Galois group of a number field for which the residual representation \(\overline{\rho }\) comes from a modular form then so does \(\rho \). This theorem has numerous hypotheses; a crucial one is that the image of \(\overline{\rho }\) must be “big,” a technical condition on subgroups of \(\mathrm {GL}_n\). In this paper we investigate this condition in compatible systems. Our main result is that in a sufficiently irreducible compatible system the residual images are big at a density one set of primes. This result should make some of the work of Clozel, Harris and Taylor easier to apply in the setting of compatible systems.

A. Snowden was partially supported by NSF fellowship DMS-0902661. A. Wiles was supported by an NSF grant.

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Notes

  1. 1.

    An argument similar to the one given here appeared in [1].

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Acknowledgements

We would like to thank Thomas Barnet-Lamb, Bhargav Bhatt, Brian Conrad, Alireza Salehi Golsefidy and Jiu-Kang Yu for useful discussions. We would also like to thank an anonymous referee for some helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    Andrew Snowden

  2. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Andrew Wiles

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  1. Andrew Snowden
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Correspondence to Andrew Wiles .

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Editors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, United Kingdom

    David Loeffler

  2. Department of Mathematics, University College London, London, United Kingdom

    Sarah Livia Zerbes

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Snowden, A., Wiles, A. (2016). Bigness in Compatible Systems. In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_13

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