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**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.

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.

Reviewer: Tamás Szamuely (Budapest)

### MSC:

14H30 | Coverings of curves, fundamental group |

11G20 | Curves over finite and local fields |

14G15 | Finite ground fields in algebraic geometry |

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\textit{M. Saïdi} and \textit{A. Tamagawa}, Algebra Number Theory 5, No. 2, 131--184 (2011; Zbl 1235.14026)

### References:

[1] | 10.2969/jmsj/00310036 · Zbl 0044.03001 |

[2] | 10.1090/S0002-9947-98-02063-7 · Zbl 0999.12004 |

[3] | 10.1006/jabr.1994.1164 · Zbl 0809.12004 |

[4] | ; Fried, Field arithmetic. Ergebnisse der Math. und ihrer Grenzgebiete (3), 11 (1986) |

[5] | ; Grothendieck, Revêtements étales et groupe fondamantal. Lecture Notes in Mathematics, 224 (1971) |

[6] | ; Harbater, Recent developments in the inverse Galois problem. Contemp. Math., 186, 353 (1995) · Zbl 0858.14013 |

[7] | ; Koenigsmann, Valuation theory and its applications. Fields Institute Commun., 33, 107 (2003) |

[8] | 10.1007/s002220050381 · Zbl 0935.14019 |

[9] | ; Mochizuki, Galois groups and fundamental groups. Math. Sci. Res. Inst. Publ., 41, 119 (2003) |

[10] | ; Mochizuki, Nagoya Math. J., 179, 17 (2005) |

[11] | ; Mochizuki, J. Math. Kyoto Univ., 47, 451 (2007) |

[12] | 10.1007/BF01425420 · Zbl 0192.40102 |

[13] | 10.1515/crll.1969.238.135 · Zbl 0201.05901 |

[14] | 10.2307/2946630 · Zbl 0814.14027 |

[15] | 10.1007/BF01241142 · Zbl 0842.14017 |

[16] | ; Rosen, Number theory in function fields. Graduate Texts in Mathematics, 210 (2002) · Zbl 1043.11079 |

[17] | 10.2977/prims/1234361157 · Zbl 1188.14016 |

[18] | ; Serre, Algebraic number theory : Proceedings of an instructional conference, 128 (1967) |

[19] | ; Serre, Cohomologie galoisienne. Lecture Notes in Math., 5 (1994) |

[20] | ; Szamuely, Séminaire Bourbaki 2002∕2003. Astérisque, 294 (2004) |

[21] | 10.1023/A:1000114400142 · Zbl 0899.14007 |

[22] | 10.2969/jmsj/02840617 · Zbl 0329.12013 |

[23] | ; Uchida, Ann. Math. (2), 106, 589 (1977) |

[24] | 10.2969/jmsj/03340595 · Zbl 0464.12004 |

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