## The inverse Galois problem and rational points on moduli spaces (1991)

Venue: | Math. Annalen |

Citations: | 56 - 26 self |

### BibTeX

@ARTICLE{Fried91theinverse,

author = {Michael D. Fried and Uc Irvine and Helmut Völklein and U Of Florida and Universität Erlangen},

title = {The inverse Galois problem and rational points on moduli spaces},

journal = {Math. Annalen},

year = {1991},

volume = {290},

pages = {771--800}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p. Many attempts have been made to realize finite groups as Galois groups of extensions of Q(x) that are regular over Q (see the end of this introduction for definitions). We call this the “regular inverse Galois problem. ” We show that to each finite group G with trivial center and integer r ≥ 3 there is canonically associated an algebraic variety, Hin r (G), defined over Q (usually reducible) satisfying the following.

### Citations

351 | Simple Groups of Lie Type - Carter - 1972 |

134 | Revetements etales et groupe fondamental - Grothendieck - 1971 |

92 |
Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten
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(Show Context)
Citation Context ...f degree n). From the above, G( ¯ k/P) =G( ¯ P/P) ∼ = ˆ Fω, and we get the exact sequence 1 → ˆ Fω → G( ¯ k/k) → ∞� Sn → 1. Moduli spaces for branched covers of P 1 were already considered by Hurwitz =-=[Hur]-=- in the special case of simple branching (where the Galois group is Sn). Fulton [Fu] showed — still in the case of simple branching — that the analytic moduli spaces studied by Hurwitz are the sets of... |

59 |
The elementary theory of .finite fields
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- 1968
(Show Context)
Citation Context ...P is a PAC-field of characteristic 0, then every finite group is regular over P. In particular, if P is also Hilbertian, then every finite group is a Galois group over P. PAC-fields first appeared in =-=[Ax]-=- and have been studied since then by various authors (cf. [FrJ]). PAC fields have projective absolute Galois group—a result of Ax [FrJ; Theorem 10.17]. Conversely, if H is a projective profinite group... |

50 |
Hurwitz schemes and irreducibility of moduli algebraic curves
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(Show Context)
Citation Context ...ence 1 → ˆ Fω → G( ¯ k/k) → ∞� Sn → 1. Moduli spaces for branched covers of P 1 were already considered by Hurwitz [Hur] in the special case of simple branching (where the Galois group is Sn). Fulton =-=[Fu]-=- showed — still in the case of simple branching — that the analytic moduli spaces studied by Hurwitz are the sets of complex points of certain schemes. Fried [Fr,1] studied more generally moduli space... |

38 | Fields of definition of function fields and Hurwitz families, Groups as Galois groups - Fried - 1977 |

29 |
The field of definition of a variety
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(Show Context)
Citation Context ...on that Aut(X/P 1 ) is trivial implies that for each β ∈ Gk1 the isomorphism δβ : X β → X with ϕ ◦ δβ = ϕ β is unique. This forces the maps δβ to satisfy Weil’s cocycle condition. By Weil’s criterion =-=[W]-=-, X can be defined over k1 such that δβ : X = X β → X is the identity for each β ∈ Gk1 . Then also ϕ : X →P1 is defined over k1. Also if ϕ is a Galois cover then it can be defined over k1 (but perhaps... |

27 | The embedding problem over an Hilbertian-PAC field
- Fried, Völklein
- 1992
(Show Context)
Citation Context ...evious one) is that every finite group is regular over the finite prime field Fp for almost all primes p (Corollary 2). In §6 we derive an addendum to our main result that is crucial for the preprint =-=[FrVo]-=-. In that paper we prove a long-standing conjecture on Hilbertian PAC-fields P (in the case char(P) = 0): Every finite embedding problem over P is solvable. For countable P this, combined with a resul... |

14 | Konstruktive Galoistheorie - Matzat - 1987 |

13 | Hurwitz families and arithmetic Galois groups - Coombes, Harbater - 1985 |

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8 |
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Citation Context ... acts transitively on Ni(C) ab or Ni(C) in . For the case of simply branched covers (i.e., C =(C,...,C) where C is the class of transpositions in G = Sn (n >2), and U = Sn−1) this was done by Clebsch =-=[Cl]-=-. In the Appendix we present a far more general result in this direction, due to Conway and Parker [CP]. §1.4. The action of Qi on E ab r and E in r : Here we indicate how the action of Qi gives formu... |

7 |
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Citation Context ...ssauer’s theorem. On the other hand, the abelian closure of any number field has projective absolute Galois group and it is Hilbertian [FrJ; Theorem 15.6]. But Frey noted that such a field isn’t PAC (=-=[Fy]-=- or [FrJ; Corollary 10.15]). Shafarevich conjectured that the abelian closure of the rationals has an ω-free absolute Galois group. Our methods yield an analogue of this (to appear in a sequel to the ... |

5 |
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Citation Context ... Ax [FrJ; Theorem 10.17]. Conversely, if H is a projective profinite group, then there exists a PAC field P such that H is the absolute Galois group of P—an observation of Lubotzky and van den Dries (=-=[LD]-=-, [FrJ; Corollary 20.16]). 12sThere are many examples of Hilbertian PAC fields inside of ¯ Q. For example, F. Pop [P] has recently announced that one obtains a PAC-field by adjoining √ −1 to the field... |

4 | Diophantine properties of subfields - Fried, Jarden - 1978 |

2 | Galois groups and - Fried - 1978 |

2 |
für die reine und angew
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- 1982
(Show Context)
Citation Context .... For example, F. Pop [P] has recently announced that one obtains a PAC-field by adjoining √ −1 to the field of all totally real algebraic numbers. This PAC-field is Hilbertian by Weissauer’s theorem =-=[Ws]-=- (which says that any proper finite extension of a Galois extension of a Hilbertian field is Hilbertian). Thus Theorem 2 applies to it. Furthermore, there are PAC-fields P with the property that they ... |

1 |
Moduli Spaces of Covers of P 1 and the Hurwitz Monodromy Group, J. für die reine und angew
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(Show Context)
Citation Context ...d by G Q ); further we assume bi �= ∞ for all i. For the moment we consider Ur only as a complex manifold. The (topological) fundamental group Hr = π1(Ur,b) is called the Hurwitz monodromy group (cf. =-=[BF]-=-). It is a quotient of π1(A r \ Dr,b) (via the map induced from the embedding of A r in P r ). The latter group is classically known to be isomorphic to the Artin braid group Br. Thus the “elementary ... |

1 |
On the Hurwitz number of arrays of group elements, unpublished preprint
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Citation Context ... 12F10, 14D20, 14E20, 14G05, 20B25, 20C25 Keywords: Riemann’s existence theorem; Galois groups; Nielsen classes; Braid and Hurwitz monodromy groups; PAC-fields. 1sUsing a theorem of Conway and Parker =-=[CP]-=- on such group actions, we conclude that the space H in r (G) has an (absolutely) irreducible component defined over Q if we allow r to be large and replace G by some group with quotient G (see §2.2).... |

1 | Regular extensions of R(x) and Rigidity - Debes, Fried |

1 | Endliche Gruppen I, Graduate Texts-Springer - Huppert - 1967 |

1 | Zöpfe und Galois’sche Gruppen, preprint - Matzat - 1990 |

1 |
The totally real numbers are PRC, preprint as of Oct. ’90
- Pop
(Show Context)
Citation Context ...H is the absolute Galois group of P—an observation of Lubotzky and van den Dries ([LD], [FrJ; Corollary 20.16]). 12sThere are many examples of Hilbertian PAC fields inside of ¯ Q. For example, F. Pop =-=[P]-=- has recently announced that one obtains a PAC-field by adjoining √ −1 to the field of all totally real algebraic numbers. This PAC-field is Hilbertian by Weissauer’s theorem [Ws] (which says that any... |

1 | Topics in Galois Theory, notes by H - Serre - 1989 |

1 |
Some finite groups which occur as Gal(L/K) where K
- Thompson
- 1984
(Show Context)
Citation Context ... relation between Hurwitz spaces and modular curves for r = 4. In the following example we determine the Q-structure of Hurwitz spaces H(C) in for which C satisfies the rigidity condition of Thompson =-=[Th]-=-. 10sExample: Hurwitz spaces and rigidity. Assume G has trivial center. The r-tuple C =(C1,...,Cr) of conjugacy classes of G is called rigid if the tuples (σ1,...,σr) ∈Er(G) with σi ∈ Ci form a single... |