## Galois representations, Hecke operators, and the $$\text{mod-}p$$ cohomology of $$\text{GL} (3,\mathbb{Z})$$ with twisted coefficients.(English)Zbl 0923.11083

The degree 3 homology of the group $$GL(3,\mathbb{Z})$$ with coefficients in the module of homogeneous polynomials of degree $$g$$ over $$\mathbb{F}_p$$ is computed for $$g\leq 200$$ and $$p\leq 541$$. From the topological realization, the homology splits into a boundary part and a quasicuspidal part. In [A. Ash, Duke Math. J. 65, 235-255 (1992; Zbl 0774.11024)] the conjecture is stated that an eigenclass under the Hecke algebra in the homology is attached to a Galois representation such that the characteristic polynomial of the Frobenius is given by the Hecke eigenvalues. In the present paper the conjecture is proven for the boundary part and explored experimentally for the quasicuspidal part.

### MSC:

 11F75 Cohomology of arithmetic groups 11F80 Galois representations

Zbl 0774.11024
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### References:

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