TY - JOUR

T1 - Weight decompositions on Étale fundamental groups

AU - Pridham, J.P.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?partnerID=yv4JPVwI&eid=2-s2.0-68149136563&md5=445cb2fc3d9382e095923139dafe0fd5

U2 - 10.1353/ajm.0.0055

DO - 10.1353/ajm.0.0055

M3 - Article

AN - SCOPUS:68149136563

SN - 0002-9327

VL - 131

SP - 869

EP - 891

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 3

ER -