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Mathematics > Number Theory

arXiv:1805.09373 (math)
[Submitted on 23 May 2018]

Title:On the cohomology of congruence subgroups of GL3 over the Eisenstein integers

Authors:Paul E. Gunnells, Mark McConnell, Dan Yasaki
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Abstract:Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices with bottom row (0,0,*) bmod n. In this paper we compute the cohomology spaces H^{nu - 1} (Gamma_0 (n); C) as a Hecke module for various levels n, where nu is the virtual cohomological dimension of Gamma. This represents the first attempt at such computations for GL_3 over an imaginary quadratic field, and complements work of Grunewald--Helling--Mennicke and Cremona, who computed the cohomology of GL_2 over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on GL_3/F.
Subjects: Number Theory (math.NT)
MSC classes: 11F75 (Primary), 11F67, 11G05, 11Y99 (Secondary)
Cite as: arXiv:1805.09373 [math.NT]
  (or arXiv:1805.09373v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1805.09373

Submission history

From: Paul E. Gunnells [view email]
[v1] Wed, 23 May 2018 18:41:40 UTC (26 KB)
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