On the Cohomology of Congruence Subgroups of GL₃ over the Eisenstein Integers

Abstract

Let F be the imaginary quadratic field of discriminant −3 and ${\mathcal{O}}_F$ its ring of integers. Let Γ be the arithmetic group ${\text{GL}}_3({\mathcal{O}}_F),$ and for any ideal $\mathfrak{n}\subset {\mathcal{O}}_F$ let ${\Gamma }_0(\mathfrak{n})$ be the congruence subgroup of level $\mathfrak{n}$ consisting of matrices with bottom row $(0,0,*)\mathrm{mod}\mathfrak{n}.$ In this paper we compute the cohomology spaces $H^{\nu -1}({\Gamma }_0(\mathfrak{n});\mathbb{C})$ as a Hecke module for various levels $\mathfrak{n},$ where ν is the virtual cohomological dimension of Γ. This represents the first attempt at such computations for GL₃ over an imaginary quadratic field, and complements work of Grunewald–Helling–Mennicke and Cremona, who computed the cohomology of ${\text{GL}}_2$ over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on ${\text{GL}}_3/F.$

Keywords:

• cohomology of arithmetic groups
• automorphic forms
• computational number theory

2010 Mathematics Subject Classification:

• Primary 11F75
• Secondary 11F67
• 11G05
• 11Y99

Notes

1 In [Gunnells 00], such points were called candidates.

2 This and other similar labels refer to the L-functions and modular forms database [The LMFDB Collaboration 13].

3 We used M. Watkins’s implementation of Hecke Grössencharacters in Magma [Bosma et al. 97] in these computations.

4 We remark that there is more to be said about this Bianchi modular form f. It is itself a base change of a weight two newform g (81.2.1.a) on ${\Gamma }_0(81)\subset {\text{SL}}_2(\mathbb{Z})$ with coefficients in the quadratic field $\mathbb{Q}(\eta )$ of discriminant 12, where ${\eta }^2=3.$ The eigenvalue of T_p, $p\nmid 3$ on g away is rational (respectively in $\mathbb{Z}·\eta$) exactly when p is inert (respectively splits) in our imaginary quadratic field F. Thus g corresponds to an abelian surface with extra twist as in [Cremona 92].

5 Strictly speaking, this does not imply that the cohomology class is cuspidal, only that it appears in the interior cohomology of Γ (see [Ash et al. 84, Harder 91] for the definition). For ${\text{GL}}_3/\mathbb{Q}$ being interior implies cuspidal. For our purposes, we will abuse notation and ignore this distinction.

Additional information

Funding

PG wishes to thank the National Science Foundation for support of this research through the NSF grant DMS-1501832. DY wishes to thank the National Security Agency for support through the NSA grant H98230-15-1-0228 and UNCG for support through the Faculty First Award.