Sums of Galois representations and arithmetic homology
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- by Avner Ash and Darrin Doud PDF
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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.References
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Additional Information
- Avner Ash
- Affiliation: Boston College, Chestnut Hill, Massachusetts 02467
- MR Author ID: 205374
- Email: Avner.Ash@bc.edu
- Darrin Doud
- Affiliation: Brigham Young University, Provo, Utah 84602
- MR Author ID: 634088
- Email: doud@math.byu.edu
- Received by editor(s): December 27, 2018
- Received by editor(s) in revised form: May 14, 2019
- Published electronically: August 14, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 273-293
- MSC (2010): Primary 11F75, 11F80
- DOI: https://doi.org/10.1090/tran/7904
- MathSciNet review: 4042875