v det([1− ρ(Frobv)q v ]|W Iv )−1, W Iv = subspace of W fixed by inertia at v. Grothendieck gave a related description of ζ∗ X using continuous p-adic representations GF → GL(H ét,c(XF ,Qp)) =: GL(W ). These are unramified almost everywhere, including at all good places away from p. Here the ith cohomology group W i above vanishes for i > 2 dimX. Grothendieck proved that if we remove the contrib...