Irreducibility of automorphic Galois representations of $GL\left(n\right)$, $n$ at most $5$

Calegari, Frank; Gee, Toby

Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1881-1912.

Nous prouvons l’irréductibilité pour $n$ inférieur ou égal à $5$ des représentations galoisiennes $l$-adiques associées aux représentations automorphes cuspidales algébriques et régulières de ${\mathrm{GL}}_n$ sur un corps totalement réel qui sont autoduales à torsion près. Nous prouvons également l’irréductibilité des représentations galoisiennes modulo $l$ pour presque tout $l$, et nous montrons l’indépendance en $l$ de l’algèbre de Lie de la clôture Zariskienne de la représentation $l$-adique.

Let $\pi$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${\mathrm{GL}}_n\left({𝔸}_F\right)$, where $F$ is a totally real field and $n$ is at most $5$. We show that for all primes $l$, the $l$-adic Galois representations associated to $\pi$ are irreducible, and for all but finitely many primes $l$, the mod $l$ Galois representations associated to $\pi$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$-adic representations are independent of $l$.

[1] Arthur, James; Clozel, Laurent Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, 120, Princeton University Press, Princeton, NJ, 1989 | MR | Zbl

[2] Asgari, Mahdi; Raghuram, A. A cuspidality criterion for the exterior square transfer of cusp forms on $\mathrm{GL}\left(4\right)$, On certain $L$-functions (Clay Math. Proc.), Volume 13, Amer. Math. Soc., Providence, RI, 2011, pp. 33-53 | MR | Zbl

[3] Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David Congruences between Hilbert modular forms: constructing ordinary lifts, Duke Mathematical Journal, Volume 161 (2012) no. 8, pp. 1521-1580 | DOI | MR

[4] Barnet-Lamb, Tom; Gee, Toby; Geraghty, David; Taylor, Richard Potential automorphy and change of weight, 2010 (preprint available at http://www.math.northwestern.edu/~gee/)

[5] Barnet-Lamb, Tom; Geraghty, David; Harris, Michael; Taylor, Richard A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci., Volume 47 (2011) no. 1, pp. 29-98 | DOI | MR | Zbl

[6] Bellaïche, Joël; Chenevier, Gaëtan The sign of Galois representations attached to automorphic forms for unitary groups, Compos. Math., Volume 147 (2011) no. 5, pp. 1337-1352 | DOI | MR | Zbl

[7] Blasius, Don; Rogawski, Jonathan D. Tate classes and arithmetic quotients of the two-ball, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 421-444 | MR | Zbl

[8] Calegari, Frank Even Galois representations and the Fontaine-Mazur conjecture, Invent. Math., Volume 185 (2011) no. 1, pp. 1-16 | DOI | MR | Zbl

[9] Calegari, Frank; Mazur, Barry Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu, Volume 8 (2009) no. 1, pp. 99-177 | DOI | MR | Zbl

[10] Clozel, Laurent; Harris, Michael; Taylor, Richard Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008) no. 108, pp. 1-181 (With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras) | DOI | Numdam | MR | Zbl

[11] Darmon, Henri; Diamond, Fred; Taylor, Richard Fermat’s last theorem, Elliptic curves, modular forms and Fermat’s last theorem (Hong Kong, 1993), Int. Press, Cambridge, MA, 1997, pp. 2-140 | MR | Zbl

[12] Dieulefait, Luis V. Uniform behavior of families of Galois representations on Siegel modular forms and the endoscopy conjecture, Bol. Soc. Mat. Mexicana (3), Volume 13 (2007) no. 2, pp. 243-253 | MR | Zbl

[13] Dieulefait, Luis V.; Vila, Núria On the classification of geometric families of four-dimensional Galois representations, Math. Res. Lett., Volume 18 (2011) no. 4, pp. 805-814 | DOI | MR

[14] Dimitrov, Mladen Galois representations modulo $p$ and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 4, pp. 505-551 | MR | Zbl

[15] Guralnick, Robert; Malle, Gunter Characteristic polynomials and fixed spaces of semisimple elements, Recent developments in Lie algebras, groups and representation theory (Proc. Sympos. Pure Math.), Volume 86, Amer. Math. Soc., Providence, RI, 2012, pp. 173-186 | MR

[16] Harder, G. Eisenstein cohomology of arithmetic groups. The case ${\mathrm{GL}}_2$, Invent. Math., Volume 89 (1987) no. 1, pp. 37-118 | DOI | MR | Zbl

[17] Harris, Michael; Taylor, Richard The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, Princeton, NJ, 2001 (With an appendix by Vladimir G. Berkovich) | MR | Zbl

[18] Jacquet, H.; Shalika, J. A. On Euler products and the classification of automorphic forms. II, Amer. J. Math., Volume 103 (1981) no. 4, pp. 777-815 | DOI | MR | Zbl

[19] Keune, Frans On the structure of the $K_2$ of the ring of integers in a number field, Proceedings of Research Symposium on $K$-Theory and its Applications (Ibadan, 1987), Volume 2 (1989), pp. 625-645 | MR | Zbl

[20] Kim, Henry H. Functoriality for the exterior square of ${\mathrm{GL}}_4$ and the symmetric fourth of ${\mathrm{GL}}_2$, J. Amer. Math. Soc., Volume 16 (2003) no. 1, p. 139-183 (electronic) (With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak) | DOI | MR | Zbl

[21] Ramakrishnan, Dinakar Modularity of solvable Artin representations of $\mathrm{GO}\left(4\right)$-type, Int. Math. Res. Not. (2002) no. 1, pp. 1-54 | DOI | MR | Zbl

[22] Ramakrishnan, Dinakar An Exercise Concerning the Self-dual Cusp Forms on $\mathrm{GL}\left(3\right)$, 2009 (preprint)

[23] Ramakrishnan, Dinakar Irreducibility of $\ell$-adic representations associated to regular cusp forms on $\mathrm{GL}\left(4\right)/\mathbb{Q}$, 2009 (preprint)

[24] Ribet, Kenneth A. Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) (Lecture Notes in Math.), Volume 601, Springer, Berlin, 1977, pp. 17-51 | MR | Zbl

[25] Taylor, Richard On Galois representations associated to Hilbert modular forms. II, Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong, 1993) (Ser. Number Theory, I), Int. Press, Cambridge, MA, 1995, pp. 185-191 | MR | Zbl

[26] Taylor, Richard The image of complex conjugation in l-adic representations associated to automorphic forms (2010) (preprint available at http://www.math.harvard.edu/ rtaylor)

[27] Taylor, Richard; Yoshida, Teruyoshi Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., Volume 20 (2007) no. 2, p. 467-493 (electronic) | DOI | MR | Zbl

[28] Wiles, A. The Iwasawa conjecture for totally real fields, Ann. of Math. (2), Volume 131 (1990) no. 3, pp. 493-540 | DOI | MR | Zbl