Semi-direct Galois covers of the affine line

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $\psi:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.Comment: minor changes in the contex

Higher Dimensional Class Field Theory: The variety case

Let k be a finite field, and suppose that the arithmetical variety X ⊂ [special characters omitted] is an open subset in projective space. Suppose that [special characters omitted] is the Wiesend idèle class group of X, [special characters omitted](X) the abelianised fundamental group, and ρ X : [special characters omitted](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 \u3c [special characters omitted] 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 ρX is injective and has dense image. We obtain a one-to-one correspondence of open geometrically bounded subgroups of [special characters omitted] with open subgroups of [special characters omitted](X). Furthermore, we show that for an étale cover X\u27\u27 → X with maximal abelian subcover X\u27 → X, the reciprocity morphism induces an isomorphism [special characters omitted] ≃ Gal(X\u27/X)