Weight decompositions on Étale fundamental groups

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Abstract / Description of output

Let X be a smooth or proper variety defined over a finite field k, and let X be the base extension of X to the algebraic closure k̄. The geometric étale fundamental group π(X, x̄) of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. For l not dividing the characteristic of k, we consider the pro-ℚ-algebraic completion of π(X, x̄) as a nonabelian 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 étale fundamental groups of smooth varieties.
Original languageEnglish
Pages (from-to)869-891
Number of pages23
JournalAmerican Journal of Mathematics
Issue number3
Publication statusPublished - 1 Jan 2009


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