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arXiv · 1609.08278

On the cohomology of $GL_3$ of elliptic curves and Quillen's conjecture

Matthias Wendt
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Abstract

The paper provides a computation of the additive structure as well as a partial description of the Chern-class module structure of the cohomology of $GL_3$ over the function ring of an elliptic curve over a finite field. The computation is achieved by a detailed analysis of the isotropy spectral sequence for the action of $GL_3$ on the associated Bruhat-Tits building. This provides insights into the function field version of Quillen's conjecture on the structure of cohomology rings of arithmetic groups. The computations exhibit a lot of explicit classes which are torsion for the Chern-class ring. In some examples, even the torsion-free quotient of cohomology fails to be free. A possible variation of Quillen's conjecture is also discussed.

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Matthias Wendt. 2016-09-27. On the cohomology of $GL_3$ of elliptic curves and Quillen's conjecture. https://arxiv.org/abs/1609.08278

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