Abelian étale coverings of generic curves and ordinarity (Q1209820)

Let \(k\) be a separably closed field of characteristic \(p>0\) and \(g \geq 2\) be an integer. By \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36(1969), 75-110 (1970; Zbl 0181.488)] there exists a universal stable curve \(Z_ g \to H_ g\), where \(H_ g\) is a \(k\)-subscheme of a convenient Hilbert scheme such that every stable curve over \(k\) of genus \(g\) is isomorphic to a fiber of \(Z_ g \to H_ g\) and \(H_ g\) is geometrically irreducible and smooth over \(k\), moreover the set of \(x \in H_ g\) whose fiber in \(Z_ g\) is smooth is an open dense subset of \(H_ g\). Let \(\eta\) be the generic point of \(H_ g\) and \(L\) the algebraic closure of \(k(\eta)\). The generic curve of genus \(g\) is denoted by \(X=Z_ g \times_{H_ g} \text{Spec} L\). It is a proper, smooth and connected curve over \(L\). Given a scheme \(S\) of characteristic \(p\) and \(f:Z \to S\) any morphism of schemes, we denote by \(Z^{(p)}=Z \times_ SS\) with respect to the absolute Frobenius morphism \(S \to S\). Given a semi-stable curve \(Z\) over a field \(K\) of characteristic \(p>0\) the relative Frobenius \(F:Z \to Z^{(p)}\) induces a map \(F^*:H^ 1(Z^{(p)}, {\mathcal O}_{Z^{(p)}}) \to H^ 1(Z, {\mathcal O}_ Z)\). We say that \(Z\) is ordinary if \(F^*\) is bijective. The author's main result states that given any étale connected Galois covering \(Y\) of \(X\) with Galois group of order prime to \(p\) then \(Y\) is ordinary. In particular, \(X\) is ordinary. Furthermore, this result together with a result of \textit{R. M. Crew} [cf. Compos. Math. 52, 31-45 (1984; Zbl 0558.14009); corollary 1.8.3] which says that if \(Y\) is a complete nonsingular connected curve defined over an algebraically closed field \(k\) of characteristic \(p>0\) and \(X \to Y\) is a finite étale Galois covering of degree a power of \(p\), then \(X\) is ordinary if and only if \(Y\) is ordinary, implies that every étale abelian covering of a generic curve is ordinary.