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\(\mathrm{SL}_3(\mathbb F_2)\)-extensions of \(\mathbb Q\) and arithmetic cohomology modulo 2. (English) Zbl 1094.11019

In [A. Ash, D. Doud and D. Pollack, Duke Math. J. 112, No. 3, 521–579 (2002; Zbl 1023.11025)] there is presented the conjecture that every mod-\(p\) Galois representation in \(\mathrm{GL}_n\) should be attached to the mod-\(p\) cohomology of the arithmetic group \(\Gamma_0 (N)\) in the sense that Frobenius eigenvalues correspond to Hecke eigenvalues. For \(n=2\) this conjecture boils down to Serre’s classical conjecture. The conjecture is tested numerically in many cases when \(p\) is an odd prime.
In the paper under review the authors test the conjecture for \(p=2\), and they offer a refinement of the conjecture resolving ambiguities in the predicted weight.

MSC:

11F75 Cohomology of arithmetic groups
11F80 Galois representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields

Citations:

Zbl 1023.11025
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References:

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