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

Frank Calegari^{[1]}; Toby Gee^{[1]}

- [1] Northwestern University Department of Mathematics 2033 Sheridan Road Evanston IL 60208-2730 (USA)

Annales de l’institut Fourier (2013)

- Volume: 63, Issue: 5, page 1881-1912
- ISSN: 0373-0956

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topCalegari, Frank, and Gee, Toby. "Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$." Annales de l’institut Fourier 63.5 (2013): 1881-1912. <http://eudml.org/doc/275430>.

@article{Calegari2013,

abstract = {Let $\pi $ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of $\textrm\{GL\}_n(\{\mathbb\{A\}\}_F)$, 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 $l$.},

affiliation = {Northwestern University Department of Mathematics 2033 Sheridan Road Evanston IL 60208-2730 (USA); Northwestern University Department of Mathematics 2033 Sheridan Road Evanston IL 60208-2730 (USA)},

author = {Calegari, Frank, Gee, Toby},

journal = {Annales de l’institut Fourier},

keywords = {Galois representations; automorphic representations; représentations galoisiennes; représentations automorphes; -adic Galois representation; cuspidal automorphic representation; totally real field},

language = {eng},

number = {5},

pages = {1881-1912},

publisher = {Association des Annales de l’institut Fourier},

title = {Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$},

url = {http://eudml.org/doc/275430},

volume = {63},

year = {2013},

}

TY - JOUR

AU - Calegari, Frank

AU - Gee, Toby

TI - Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$

JO - Annales de l’institut Fourier

PY - 2013

PB - Association des Annales de l’institut Fourier

VL - 63

IS - 5

SP - 1881

EP - 1912

AB - Let $\pi $ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of $\textrm{GL}_n({\mathbb{A}}_F)$, 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 $l$.

LA - eng

KW - Galois representations; automorphic representations; représentations galoisiennes; représentations automorphes; -adic Galois representation; cuspidal automorphic representation; totally real field

UR - http://eudml.org/doc/275430

ER -

## References

top- James Arthur, Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, 120 (1989), Princeton University Press, Princeton, NJ Zbl0682.10022MR1007299
- Mahdi Asgari, A. Raghuram, A cuspidality criterion for the exterior square transfer of cusp forms on $\mathrm{GL}\left(4\right)$, On certain -functions 13 (2011), 33-53, Amer. Math. Soc., Providence, RI Zbl1275.11086MR2767509
- Thomas Barnet-Lamb, Toby Gee, David Geraghty, Congruences between Hilbert modular forms: constructing ordinary lifts, Duke Mathematical Journal 161 (2012), 1521-1580 Zbl1297.11028MR2931274
- Tom Barnet-Lamb, Toby Gee, David Geraghty, Richard Taylor, Potential automorphy and change of weight, (2010) Zbl1310.11060
- Tom Barnet-Lamb, David Geraghty, Michael Harris, Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), 29-98 Zbl1264.11044MR2827723
- Joël Bellaïche, Gaëtan Chenevier, The sign of Galois representations attached to automorphic forms for unitary groups, Compos. Math. 147 (2011), 1337-1352 Zbl1259.11058MR2834723
- Don Blasius, Jonathan D. Rogawski, Tate classes and arithmetic quotients of the two-ball, The zeta functions of Picard modular surfaces (1992), 421-444, Univ. Montréal, Montreal, QC Zbl0828.14012MR1155236
- Frank Calegari, Even Galois representations and the Fontaine-Mazur conjecture, Invent. Math. 185 (2011), 1-16 Zbl1231.11058MR2810794
- Frank Calegari, Barry Mazur, Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu 8 (2009), 99-177 Zbl1211.11065MR2461903
- Laurent Clozel, Michael Harris, Richard Taylor, Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008), 1-181 Zbl1169.11020MR2470687
- Henri Darmon, Fred Diamond, Richard Taylor, Fermat’s last theorem, Elliptic curves, modular forms and Fermat’s last theorem (Hong Kong, 1993) (1997), 2-140, Int. Press, Cambridge, MA Zbl0877.11035MR1605752
- Luis V. Dieulefait, Uniform behavior of families of Galois representations on Siegel modular forms and the endoscopy conjecture, Bol. Soc. Mat. Mexicana (3) 13 (2007), 243-253 Zbl1173.11032MR2472504
- Luis V. Dieulefait, Núria Vila, On the classification of geometric families of four-dimensional Galois representations, Math. Res. Lett. 18 (2011), 805-814 Zbl1296.11051MR2831844
- Mladen Dimitrov, Galois representations modulo $p$ and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. (4) 38 (2005), 505-551 Zbl1160.11325MR2172950
- Robert Guralnick, Gunter Malle, Characteristic polynomials and fixed spaces of semisimple elements, Recent developments in Lie algebras, groups and representation theory 86 (2012), 173-186, Amer. Math. Soc., Providence, RI Zbl1320.20042MR2977003
- G. Harder, Eisenstein cohomology of arithmetic groups. The case ${\mathrm{GL}}_{2}$, Invent. Math. 89 (1987), 37-118 Zbl0629.10023MR892187
- Michael Harris, Richard Taylor, The geometry and cohomology of some simple Shimura varieties, 151 (2001), Princeton University Press, Princeton, NJ Zbl1036.11027MR1876802
- H. Jacquet, J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), 777-815 Zbl0491.10020MR623137
- Frans Keune, On the structure of the ${K}_{2}$ of the ring of integers in a number field, Proceedings of Research Symposium on -Theory and its Applications (Ibadan, 1987) 2 (1989), 625-645 Zbl0705.19007MR999397
- Henry H. Kim, Functoriality for the exterior square of ${\mathrm{GL}}_{4}$ and the symmetric fourth of ${\mathrm{GL}}_{2}$, J. Amer. Math. Soc. 16 (2003), 139-183 (electronic) Zbl1018.11024MR1937203
- Dinakar Ramakrishnan, Modularity of solvable Artin representations of $\mathrm{GO}\left(4\right)$-type, Int. Math. Res. Not. (2002), 1-54 Zbl1002.11045MR1874921
- Dinakar Ramakrishnan, An Exercise Concerning the Self-dual Cusp Forms on $\mathrm{GL}\left(3\right)$, (2009) Zbl1262.11064
- Dinakar Ramakrishnan, Irreducibility of $\ell $-adic representations associated to regular cusp forms on $\mathrm{GL}\left(4\right)/\mathbb{Q}$, (2009) Zbl1262.11064
- Kenneth A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) 601 (1977), 17-51, Springer, Berlin Zbl0363.10015MR453647
- Richard Taylor, On Galois representations associated to Hilbert modular forms. II, Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong, 1993) (1995), 185-191, Int. Press, Cambridge, MA Zbl0836.11017MR1363502
- Richard Taylor, The image of complex conjugation in l-adic representations associated to automorphic forms, (2010) Zbl1303.11065
- Richard Taylor, Teruyoshi Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), 467-493 (electronic) Zbl1210.11118MR2276777
- A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493-540 Zbl0719.11071MR1053488

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