Reducible Galois representations and arithmetic homology for $\protect \mathrm{GL}(4)$

Avner Ash; Darrin Doud

doi:10.5802/ambp.375

Reducible Galois representations and arithmetic homology for

GL (4)

Avner Ash ¹ ; Darrin Doud ²

¹ Boston College Chestnut Hill MA 02467, USA
² Brigham Young University Provo UT 84602, USA

Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 2, pp. 207-246.

Résumé

We prove that a sum of two odd irreducible two-dimensional Galois representations with squarefree relatively prime Serre conductors is attached to a Hecke eigenclass in the homology of a subgroup of $GL (4, ℤ)$ , with the level, nebentype, and coefficient module of the homology predicted by a generalization of Serre’s conjecture to higher dimensions. To do this we prove along the way that any Hecke eigenclass in the homology of a congruence subgroup of a maximal parabolic subgroup of $GL (n, ℚ)$ has a reducible Galois representation attached, where the dimensions of the components correspond to the type of the parabolic subgroup. Our main new tool is a resolution of $ℤ$ by $GL (n, ℚ)$ -modules consisting of sums of Steinberg modules for all subspaces of $ℚ^{n}$ .

Publié le : 2018-11-27

DOI : 10.5802/ambp.375

Classification : 11F75, 11R80
Mots clés : Galois representations, arithmetic homology

Affiliations des auteurs :

Avner Ash ¹ ; Darrin Doud ²

¹ Boston College Chestnut Hill MA 02467, USA
² Brigham Young University Provo UT 84602, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Avner Ash and Darrin Doud},
     title = {Reducible {Galois} representations and arithmetic homology for $\protect \mathrm{GL}(4)$},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {207--246},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {25},
     number = {2},
     year = {2018},
     doi = {10.5802/ambp.375},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.375/}
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Avner Ash; Darrin Doud. Reducible Galois representations and arithmetic homology for $\protect \mathrm{GL}(4)$. Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 2, pp. 207-246. doi : 10.5802/ambp.375. https://ambp.centre-mersenne.org/articles/10.5802/ambp.375/

Bibliographie
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