On the semisimplicity conjecture and Galois representations

Fu, Lei

doi:10.1090/S0002-9947-01-02814-8

On the semisimplicity conjecture and Galois representations
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by Lei Fu

Trans. Amer. Math. Soc. 353 (2001), 4357-4369

DOI: https://doi.org/10.1090/S0002-9947-01-02814-8

Published electronically: June 21, 2001

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Abstract:

The semisimplicity conjecture says that for any smooth projective scheme $X_0$ over a finite field $\mathbf {F}_q$, the Frobenius correspondence acts semisimply on $H^i(X\otimes _{\mathbf { F}_q} \mathbf { F}, \overline {\mathbf { Q}}_l)$, where $\mathbf { F}$ is an algebraic closure of $\mathbf { F}_q$. Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).

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