# Irreducibility of automorphic Galois representations of low dimensions

Article

First Online:

- 50 Downloads

## Abstract

Let \(\pi \) be a polarizable, regular algebraic, cuspidal automorphic representation of \(\text { GL }_n(\mathbb {A}_F)\), where *F* is an imaginary CM field and \(n \le 6\). We show that there is a Dirichlet density 1 set \(\mathfrak {L}\) of rational primes, such that for all \(l\in \mathfrak {L}\), the *l*-adic Galois representations associated to \(\pi \) are irreducible.

## Mathematics Subject Classification

11F80 11F22 11F70## Notes

### Acknowledgements

I would like to thank my Ph.D. advisor Richard Taylor for proposing this problem to me and for many helpful discussions during the completion of this manuscript.

## References

- 1.Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math.
**179**, 501–609 (2014)MathSciNetCrossRefGoogle Scholar - 2.Burde, D., Moens, W.: Minimal faithful representations of reductive Lie algebras. Arch. Math.
**89**(6), 513–523 (2007)MathSciNetCrossRefGoogle Scholar - 3.Calegari, F., Gee, T.: Irreducibility of automorphic Galois representations of \(\text{ GL }(n)\), \(n\) at most 5. Ann. l’Inst. Fourier
**63**(5), 1881–1912 (2013)MathSciNetCrossRefGoogle Scholar - 4.Caraiani, A.: Monodromy and local-global compatibility for \(l=p\). Algebra Number Theory
**8**(7), 1597–1646 (2014)MathSciNetCrossRefGoogle Scholar - 5.Henniart, G.: Représentations l-adiques abéliennes, Progr. Math., vol. 22, pp. 107–126. Birkhäuser, Boston (1982)Google Scholar
- 6.Hui, C.: Monodromy of Galois representations and equal-rank subalgebra equivalence. Math. Res. Lett.
**20**(4), 1–24 (2013)MathSciNetCrossRefGoogle Scholar - 7.Humphreys, J.: Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21. Springer, New York (1975)CrossRefGoogle Scholar
- 8.Larsen, M., Pink, R.: On \(l\)-independence of algebraic monodromy groups in compatible systems of representations. Invent. Math.
**107**(3), 603–636 (1992)MathSciNetCrossRefGoogle Scholar - 9.Patrikis, S.: Variations on a theorem of Tate. ArXiv e-prints (2012)Google Scholar
- 10.Patrikis, S., Taylor, R.: Automorphy and irreducibility of some \(l\)-adic representations. Compos. Math.
**151**, 207–229 (2015)MathSciNetCrossRefGoogle Scholar - 11.Sen, S.: Lie algebras of Galois groups arising from Hodge–Tate modules. Ann. Math.
**97**, 160–170 (1973)MathSciNetCrossRefGoogle Scholar - 12.Serre, J.: Letter to Ribet (1981)Google Scholar
- 13.Serre, J.: Abelian \(l\)-adic Representations and Elliptic Curves, Research Notes in Mathematics, vol. 7. Addison-Wesley, London (1998)zbMATHGoogle Scholar
- 14.Springer, T.A.: Reductive groups. In: Borel, A., Casselman, W. (eds.) Automorphic Forms, Representations and \(L\)-Functions, Part 1. Proceedings of Symposia in Pure Mathematics, vol. 33.1, pp. 3–28 (1979)Google Scholar
- 15.Springer, T.: Twisted conjugacy in simply connected groups. Transform. Groups
**11**(3), 539–545 (2006). https://doi.org/10.1007/s00031-005-1113-6 MathSciNetCrossRefzbMATHGoogle Scholar - 16.Steinberg, R.: Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, vol. 80. American Mathematical Society, Providence (1968)Google Scholar

## Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018