Fields of definition of p-adic covers

Dèbes, Pierre; Harbater, David

doi:10.1515/crll.1998.052

Cited by 6 publications

(9 citation statements)

References 5 publications

(22 reference statements)

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“…The last corollary conjoins Corollary 1.3 above and results of Dèbes-Harbater [7] and Emsalem [9]. We assume here that the base space B is a curve.…”

Section: Applicationssupporting

confidence: 54%

“…It can be shown then that if K is the field of moduli of the (G-)cover f , then K v is a field of definition of f for all good places v of K; the result was proved for G-covers of P 1 in [7] and generalized in [9]. We obtain Corollary 1.4.…”

Section: Applicationsmentioning

confidence: 72%

“…KðvÞ corresponds to the field extension obtained by specializing the coe‰cients of the ''generic'' polynomial 8) For Q-covers, it is tempting to use the following alternate argument: the field of moduli K is the intersection of all fields of definition ( [2], §3.4.1, [7]); therefore, from assertion (2) of Main Theorem A (whose proof given below does not use assertion (1)), K contains the intersection of all fields of definition of points in V. However one cannot conclude that K is a field of definition of V : in the preceding sentence, fields of definition of points should be understood as fields of definition of coordinates in an a‰ne model; for points in a proper Zariski closed subset, this may not coincide with the intrinsic notion of fields of definition of points on a scheme (think of the a‰ne line y ¼ ffiffi ffi 2 p x which is defined over Qð ffiffi ffi 2 p Þ but passes through ð0; 0Þ).…”

Section: Proof Of Main Theorem a (2)mentioning

confidence: 99%

See 2 more Smart Citations

Descent varieties for algebraic covers

Dèbes¹,

Douai²,

Moret-Bailly³

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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This paper is about descent theory for algebraic covers. Typical questions concern fields of definition, models, moduli spaces, families, etc. of covers. Here we construct descent varieties. Associated to any given cover f , they have the property that whether they have rational points on a given field k is the obstruction to descending the field of definition of f to k. Our constructions have a global version above moduli spaces of covers (Hurwitz spaces): here descent varieties are parameter spaces for Hurwitz families with some versal property. Descent varieties provide a new diophantine viewpoint on descent theory by reducing the questions to that of finding points on varieties. There are concrete applications. We answer a question raised in [7] about totally p-adic models of covers. We also show that the subset of a given moduli space of covers of P 1 where the field of moduli is a field of definition is Zariski-dense.

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“…The last corollary conjoins Corollary 1.3 above and results of Dèbes-Harbater [7] and Emsalem [9]. We assume here that the base space B is a curve.…”

Section: Applicationssupporting

confidence: 54%

Section: Applicationsmentioning

confidence: 72%

Section: Proof Of Main Theorem a (2)mentioning

confidence: 99%

See 1 more Smart Citation

Descent varieties for algebraic covers

Dèbes¹,

Douai²,

Moret-Bailly³

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In this direction the following can be obtained from [7] and [10]. The same holds if the residue characteristic does not divide the order of Aut(X) and if the branch locus of the cover X → X/Aut(X) is smooth at v. From results of Fulton [12], these assumptions ensure that X has k ur v -good reduction (a fortiori k ur v -stable reduction) with respect to k v .…”

Section: 4mentioning

confidence: 95%

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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“…Also, under (GR0), the field of moduli of a G-cover with general base B is a field of definition (see [Ems99], which generalizes [DH98]). Hence theorem 2.1 is the special case of theorem 2.6 with B = P 1 and the good reduction assumption ensured by (GR0).…”

Section: Torsion Of Abelian Varietiesmentioning

confidence: 99%

Abelian constraints in inverse Galois theory

Cadoret

Dèbes

2008

manuscripta math.

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Abstract. We show that if a finite group G is the Galois group of a Galois cover of P 1 over Q, then the orders p n of the abelianization of its p-Sylow subgroups are bounded in terms of their index m, of the branch point number r and the smallest prime | |G| of good reduction of the branch divisor. This is a new constraint for the regular inverse Galois problem: if p n is suitably large compared to r and m, the branch points must coalesce modulo small primes. We further conjecture that p n should be bounded only in terms of r and m. We use a connection with some rationality question on the torsion of abelian varieties. For example, our conjecture follows from the so-called torsion conjectures. Our approach also yields a new viewpoint on Fried's Modular Tower program.

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Fields of definition of p-adic covers

Cited by 6 publications

References 5 publications

Descent varieties for algebraic covers

Descent varieties for algebraic covers

Finiteness Results in Descent Theory

Abelian constraints in inverse Galois theory

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