## On the Hom-form of Grothendieck’s birational anabelian conjecture in positive characteristic.(English)Zbl 1235.14026

The authors continue their investigation of the birational abelian conjecture for function fields of curves over finite fields started in [Publ. Res. Inst. Math. Sci. 45, No. 1, 135–186 (2009; Zbl 1188.14016)]. The aim is to recover finite separable extensions of such function fields from open homomorphisms between their absolute Galois groups (or better, from geometrically prime-to-$$p$$ quotients of these). Contrary to the now well-understood case of isomorphisms of Galois groups, the main obstacle here is the current lack of a satisfactory local theory for decomposition and inertia groups. So roughly the authors show that this is the only obstacle: if a satisfactory local theory exists, then the Galois characterization of finite separable extensions of function fields is possible.
More precisely, the authors call a continuous open homomorphism $$\phi: G_1\to G_2$$ between absolute Galois groups of function fields of curves over finite fields rigid if, up to replacing $$G_1$$ and $$G_2$$ by suitable open subgroups, $$\phi$$ maps each decomposition group of a closed point in $$G_1$$ to a decomposition group in $$G_1$$. The first main result is that, up to composing by inner automorphisms, rigid homomorphisms correspond bijectively to finite separable extensions of function fields. If instead of the full Galois group of a function field $$K$$ over a finite field $$k$$ of characteristic $$p$$ one considers the extension of the absolute Galois group of $$k$$ by the maximal prime-to-$$p$$ quotient of that of $$K{\bar k}$$, it is possible to recover from rigid homomorphisms only those extensions that become of degree prime to $$p$$ after base change to the algebraic closure of the base field. In order to recover finite separable extensions in this case as well, the authors impose a more involved technical condition on homomorphisms of Galois groups that they call proper and inertia-rigid.
The (technically difficult) proofs are refinements of the arguments of earlier papers, in particular those of [loc. cit.]. In the case of rigid homomorphisms such a refinement is possible but the authors are able to give a quicker proof by reducing the statement to the case of isomorphisms treated by K. Uchida [Ann. Math. (2) 106, 589–598 (1977; Zbl 0372.12017)] for the full Galois group and in their previous paper in the prime-to-$$p$$ case.

### MSC:

 14H30 Coverings of curves, fundamental group 11G20 Curves over finite and local fields 14G15 Finite ground fields in algebraic geometry

### Citations:

Zbl 1188.14016; Zbl 0372.12017
Full Text:

### References:

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