## Abelian étale coverings of generic curves and ordinarity. (Revêtements étales abéliens de courbes génériques et ordinarité.)(French)Zbl 0785.14014

Let $$k$$ be a separably closed field of characteristic $$p>0$$ and $$g \geq 2$$ be an integer. By P. Deligne and 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 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.

### MSC:

 14G15 Finite ground fields in algebraic geometry 14H30 Coverings of curves, fundamental group

### Citations:

Zbl 0181.488; Zbl 0558.14009
Full Text:

### References:

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