| Let k be a finite field, and suppose that the arithmetical variety X ⊂ Pnk is an open subset in projective space. Suppose that CX is the Wiesend idele class group of X, pab1 (X) the abelianised fundamental group, and rho X : CX&rarrr;pab 1 (X) the Wiesend reciprocity map. We use the Artin-Schreier-Witt and Kummer Theory of affine k-algebras to prove a full reciprocity law for X. We find necessary and sufficent conditions for a subgroup H < CX to be a norm subgroup: H is a norm subgroup if and only if it is open and its induced covering datum is geometrically bounded. We show that rhoX is injective and has dense image. We obtain a one-to-one correspondence of open geometrically bounded subgroups of CX with open subgroups of pab1 (X). Furthermore, we show that for an etale cover X'' → X with maximal abelian subcover X' → X, the reciprocity morphism induces an isomorphism CX/NCX' ' ≃ Gal(X'/X). |
