Everywhere unramified automorphic cohomology for \(\mathrm{SL}_3(\mathbb Z)\). (English) Zbl 1154.11019

The authors consider automorphic cohomology for \(\text{SL}_3(\mathbb{Z})\) defined over \(\mathbb{Q}\) and unramified everywhere, i.e. \(H^*_{\text{cusp}}(\text{SL}_n(\mathbb{Z}), V)= 0\) for all irreducible finite-dimensional complex rational representations \(V\) of \(\text{GL}_n(\mathbb{C})\). Certain instances of Langlands functoriality (for instance, the symmetric square lift) would lift everywhere unramified \(\text{GL}_2\)-representations to \(\text{GL}_3\)-representations, again everywhere unramified.
The authors conjecture that the only irreducible cuspidal automorphic representation for \(\text{GL}_3/\mathbb{Q}\) of cohomological type and level one are the symmetric square lifts of classical cuspforms of level one (Conjecture 1.1 and Remarks 1–5). The computational evidence for this conjecture is presented for the irreducible representations \(W_g\) of \(\text{GL}_3(\mathbb{C})\) of highest weight \((2g,g,0)\), up to \(g= 120\).


11F75 Cohomology of arithmetic groups
11F55 Other groups and their modular and automorphic forms (several variables)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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