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On the cohomology of congruence subgroups of \(\mathrm{GL}_3\) over the Eisenstein integers. (English) Zbl 1485.11096

The authors calculate cohomology with trivial complex coefficients of certain congruence subgroups in \(\mathrm{GL}(3,\mathcal{O}_F)\) in degree 5, where \(\mathcal{O}_F\) is the ring of Eisenstein integers with quotient field \(F.\) They also compute the action of finitely many Hecke operators on this cohomology.
In many cases of their calculations they can also describe an object of arithmetic geometric origin (or of motivic nature) such that the Galois action there matches with the Hecke operation on cohomology as it is expected and conjectured in general, thereby giving evidence for this general conjecture. In particular, there are examples from the calculations which seem to correspond to nonselfdual automorphic forms over \(F.\)

MSC:

11F75 Cohomology of arithmetic groups
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
11Y99 Computational number theory
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