Weight decompositions on etale fundamental groups

Details Weight decompositions on etale fundamental groups Weight decompositions on etale fundamental groups

2005-10-12

arxiv.org/abs/math/0510245v5

Amer. J. Math. 131 (2009), no. 3, 869-891

Mathematics

Algebraic Geometry

19 pages; v5 proofs greatly simplified - middle sections removed

Scientific paper

Let X be a smooth or proper variety defined over a finite field. The geometric etale fundamental group of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Q_l-algebraic completion of the fundamental group as a non-abelian Weil representation. Lafforgue's Theorem and Deligne's Weil II theorems imply that this affine group scheme is mixed, in the sense that its structure sheaf is a mixed Weil representation. When X is smooth, weight restrictions apply, affecting the possibilities for the structure of this group. This gives new examples of groups which cannot arise as etale fundamental groups of smooth varieties.

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