Finite Quotients of the Algebraic Fundamental Group of Projective Curves in Positive Characteristic

Let X be a smooth projective connected algebraic curve of genus g defined over an algebraically closed field k of characteristic p > 0. In this paper we study necessary and sufficient conditions for a finite group G to be a quotient of the algebraic fundamental group π1(X) of X. We denote by πA(X) the set of isomorphism classes of finite groups which are quotients of π1(X). Recall that a group G ∈ πA(X) will occur as a Galois group of an étale Galois cover Z → X. In this paper we will call Z → X a Galois G-cover. Let G be a finite group and suppose that its order is not divisible by p. In [Groth71, Corollary 2.12] Grothendieck showed that G ∈ πA(X) if and only if G is a quotient of the topological fundamental group Γg of a compact Riemann surface of genus g. We consider next a finite p-group G. Denote by Φ(G) = [G,G]Gp its Frattini subgroup and let G = G/Φ(G). This group is an elementary pabelian group. The p-torsion subgroup JX [p] of the Jacobian variety JX of X is an Fp-vector space whose dimension γX is called the Hasse-Witt invariant of X. It follows from [Ser56, §11] that G ∈ πA(X) if and only if G has p-rank at most γX . Suppose now that G ∈ πA(X), then G ∈ πA(X), therefore the p-rank of G (the minimal number of generators of its maximal p-quotient) is at most γX . Actually, this condition is also sufficient. This follows from the fact that the p-cohomological dimension cdp(π1(X)) of π1(X) is at most 1 (cf. end of proof of Theorem 1.3). Now these two situations are understood, the next step to study is the case of a finite group G whose order is divisible by p. Consider the case where G has a normal p-Sylow subgroup P . Let H = G/P . The main