Local-global principles for algebraic covers

Dèbes, Pierre; Douai, Jean-Claude

doi:10.1007/bf02762275

Cited by 11 publications

(9 citation statements)

References 7 publications

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“…This "local-to-global principle" was later proven to hold for "most" number fields F , including Q, in [De], Theorem 7.1 (with the possible exceptional F 's corresponding to the special case of the Grunwald-Wang Theorem). In fact, it is even true if "all F v " is replaced by "all but possibly one F v " ( [DeDo2], §3.4). Moreover, without restriction on the number field, F is the absolute field of moduli (over F ) if and only if all but finitely many completions F v are fields of definition (the "global-to-local principle" of [De], Theorem 8.1).…”

Section: §1: Introductionmentioning

confidence: 99%

“…Meanwhile, the completion of this cover at any finite place p has field of moduli Q p . If all of these p-adic completions were defined over their fields of moduli, then the field of moduli would be a field of definition at all but one place (the infinite completion); and thus [DeDo2], §3.4, would imply that Q would be a field of definition -a contradiction.…”

Section: §1: Introductionmentioning

confidence: 99%

“…By hypothesis, this is also the case for p ∈ S bad , and hence for all finite primes p. Since there is only one infinite prime, and since the special case of Grunwald-Wang does not include Q (cf. [DeDo2], §3.2), we may apply the local-to-global principle in [DeDo2] (see Theorem 3.7(b) and §3.4) to conclude that Q is a field of definition of the G-Galois cover f .…”

Section: §1: Introductionmentioning

confidence: 99%

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Fields of definition of p-adic covers

Dèbes¹,

Harbater²

1998

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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This paper concerns fields of definition and fields of moduli of G-Galois covers of the line over p-adic fields, and more generally over henselian discrete valuation fields. We show that the field of moduli of a p-adic cover will be a field of definition provided that the residue characteristic p does not divide |G| and that the branch points do not coalesce modulo p (or in the more general case, that the branch locus is smooth on the special fibre). Hence if p does not divide |G|, then a G-Galois cover of the Q-line with field of moduli Q will be defined over a number field contained in Q p if the branch points do not coalesce modulo p. This provides an explicit global-to-local principle for p-adic covers.

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Section: §1: Introductionmentioning

confidence: 99%

Section: §1: Introductionmentioning

confidence: 99%

Section: §1: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fields of definition of p-adic covers

Dèbes¹,

Harbater²

1998

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Remark 2.3. The full strength of the local-global principle for G-covers (as in [4]) is not used here; as m-roots of unity have been adjoined to K, this reduces to injectivity of the local-global map Br(L) → ⊕ w Br(L w ) (see [5,Proposition 3.4)].…”

Section: 6mentioning

confidence: 99%

Finiteness Results in Descent Theory

Dèbes

Derome

2003

J. London Math. Soc.

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It is shown that a Q-curve of genus g and with stable reduction (in some generalized sense) at every finite place outside a finite set S can be defined over a finite extension L of its field of moduli K depending only on g, S and K. Furthermore, there exist L-models that inherit all places of good and stable reduction of the original curve (except possibly for finitely many exceptional places depending on g, K and S). This descent result yields this moduli form of the Shafarevich conjecture: given g, K and S as above, only finitely many K-points on the moduli space M g correspond to Q-curves of genus g and with good reduction outside S. Other applications to arithmetic geometry, like a modular generalization of the Mordell conjecture, are given.

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“…For the theory of gerbes, which is a classical alternative to Galois cohomology for descent problems ( [9], [10]) we refer to [18] and [11].…”

Section: Proposition 24 (Cohomological Obstruction)mentioning

confidence: 99%

Lifting results for rational points on Hurwitz moduli spaces

Cadoret

2008

Isr. J. Math.

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Abstract. Hurwitz moduli spaces for G-covers of the projective line have two classical variants whether Gcovers are considered modulo the action of PGL 2 on the base or not. A central result of this paper is that, given an integer r ≥ 3 there exists a bound d(r) ≥ 1 depending only on r such that any rational point p rd of a reduced (i.e. modulo PGL 2 ) Hurwitz space can be lifted to a rational point p on the non reduced Hurwitz space with [κ(p) : κ(p rd )] ≤ d(r). This result can also be generalized to infinite towers of Hurwitz spaces. Introducing a new Galois invariant for G-covers, which we call the base invariant, we improve this result for G-covers with a non trivial base invariant. For the sublocus corresponding to such G-covers the bound d(r) can be chosen depending only on the base invariant (no longer on r) and ≤ 6. When r = 4, our method can still be refined to provide effective criteria to lift k-rational points from reduced to non reduced Hurwitz spaces. This, in particular, leads to a rigidity criterion, a genus 0 method and, what we call an expansion method to realize finite groups as regular Galois groups over Q. Some specific examples are given.

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Local-global principles for algebraic covers

Cited by 11 publications

References 7 publications

Fields of definition of p-adic covers

Fields of definition of p-adic covers

Finiteness Results in Descent Theory

Lifting results for rational points on Hurwitz moduli spaces

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