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IMAGES OF ADELIC GALOIS REPRESENTATIONS FOR MODULAR FORMS

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  03 August 2016

DAVID LOEFFLER
DAVID LOEFFLER*
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK e-mail: d.a.loeffler@warwick.ac.uk
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Abstract

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We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

MSC classification

Primary: 11F80: Galois representations
Secondary: 11R56: Adèle rings and groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 11 - 25
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Ghate, E., González-Jiménez, E. and Quer, J., On the Brauer class of modular endomorphism algebras, Int. Math. Res. Notices 2005 (12) (2005), 701–723.Google Scholar
2. Kings, G., Loeffler, D. and Zerbes, S. L., Rankin–Eisenstein classes and explicit reciprocity laws, preprint, (2015), arXiv:1503.02888.Google Scholar
3. Lei, A., Loeffler, D. and Zerbes, S. L., Euler systems for Rankin–Selberg convolutions of modular forms, Ann. of Math. 180 (2) (2014), 653–771.CrossRefGoogle Scholar
4. Momose, F., On the ℓ-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1) (1981), 89–109.Google Scholar
5. Nekovář, J., Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two, Canad. J. Math. 64 (3) (2012), 588–668.CrossRefGoogle Scholar
6. Ramakrishnan, D., Recovering modular forms from squares. Appendix to A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations (by Duke, W. and Kowalski, E.), Invent. Math. 139 (1) (2000), 1–39.Google Scholar
7. Ribet, K. A., On l-adic representations attached to modular forms, Invent. Math. 28 (1975), 245–275.CrossRefGoogle Scholar
8. Ribet, K. A., On l-adic representations attached to modular forms. II, Glasgow Math. J. 27 (1985), 185–194.Google Scholar
9. Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
10. Serre, J.-P., Abelian l-adic representations and elliptic curves, McGill University Lecture Notes written with the collaboration of Willem Kuyk and John Labute (W. A. Benjamin, Inc., New York–Amsterdam, 1968).Google Scholar
11. Serre, J.-P., Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (4) (1972), 259–331.CrossRefGoogle Scholar
12. Serre, J.-P., Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55 (Amer. Math. Soc., Providence, RI, 1994), 377–400.Google Scholar
13. Suzuki, M., Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 247 (Springer, Berlin, 1982), Translated from the Japanese by the author.CrossRefGoogle Scholar