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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sums of Galois representations and arithmetic homology


Authors: Avner Ash and Darrin Doud
Journal: Trans. Amer. Math. Soc. 373 (2020), 273-293
MSC (2010): Primary 11F75, 11F80
DOI: https://doi.org/10.1090/tran/7904
Published electronically: August 14, 2019
MathSciNet review: 4042875
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Abstract: Let $ \Gamma _0(n,N)$ denote the usual congruence subgroup of type $ \Gamma _0$ and level $ N$ in $ \text {SL}(n,{\mathbb{Z}})$. Suppose for $ i=1,2$ that we have an irreducible odd$ n$-dimensional Galois representation $ \rho _i$ attached to a homology Hecke eigenclass in $ H_*(\Gamma _0(n,N_i),M_i)$, where the level $ N_i$ and the weight and nebentype making up $ M_i$ are as predicted by the Serre-style conjecture of Ash, Doud, Pollack, and Sinnott. We assume that $ n$ is odd, that $ N_1N_2$ is squarefree, and that $ \rho _1\oplus \rho _2$ is odd. We prove two theorems that assert that $ \rho _1\oplus \rho _2$ is attached to a homology Hecke eigenclass in $ H_*(\Gamma _0(2n,N),M)$, where $ N$ and $ M$ are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of $ M_1$ and $ M_2$ are small in a certain sense. The second theorem requires the truth of a conjecture as to what degrees of homology can support Hecke eigenclasses with irreducible Galois representations attached, but no hypothesis on the highest weights of $ M_1$ and
$ M_2$. This conjecture is known to be true for $ n=3$, so we obtain unconditional results for $ \text {GL}(6)$. A similar result for $ \text {GL}(4)$ appeared in an earlier paper.


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Additional Information

Avner Ash
Affiliation: Boston College, Chestnut Hill, Massachusetts 02467
Email: Avner.Ash@bc.edu

Darrin Doud
Affiliation: Brigham Young University, Provo, Utah 84602
Email: doud@math.byu.edu

DOI: https://doi.org/10.1090/tran/7904
Received by editor(s): December 27, 2018
Received by editor(s) in revised form: May 14, 2019
Published electronically: August 14, 2019
Article copyright: © Copyright 2019 American Mathematical Society