On fields of moduli of curves

Dèbes, Pierre; Emsalem, Michel

doi:10.1006/jabr.1998.7586

Cited by 44 publications

(68 citation statements)

References 7 publications

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“…By ideas from [8] (see also [10] and Theorem 5 in [35]), we have Proposition 1. The fields of moduli of regular dessins and of quasiplatonic curves are their respective minimal fields of definition.…”

Section: Known Methods and Their Variantsmentioning

confidence: 94%

Galois actions on regular dessins of small genera

Conder¹,

Jones²,

Streit³

et al. 2013

Rev. Mat. Iberoam.

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Abstract. Dessins d'enfants can be regarded as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve defined over a number field). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge-transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the fields of definition for all curves with regular dessins of genus 2 to 18. The many facets of dessinsOne may introduce dessins d'enfants as hypermaps on compact oriented 2-manifolds. These objects, first studied in genus 0 by Cori in [9], have several equivalent algebraic or topological definitions. Topologically they are a generalisation of maps, whose underlying graphs are replaced with hypergraphs, in which hyperedges are allowed to be incident with any finite number of hypervertices and hyperfaces. One way to visualise this concept is to represent a hypermap D by its Walsh map W (D) , a connected bipartite graph B embedded in a compact oriented surface, dividing it into simply connected cells [31]. In the language of hypermaps, the white and black vertices represent the hypervertices and hyperedges of D, the edges represent incidences between them, and the cells represent the hyperfaces.There are several other ways to define dessins. There is a group theoretic description of maps and hypermaps by their (hyper)cartographic groups; see, for example, [18], [19] or the introduction of [20]. The latter are the monodromy groups of the functions about to be introduced, and give a link to function theory on Riemann surfaces and algebraic geometry on curves. Playing a key role are

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Section: Known Methods and Their Variantsmentioning

confidence: 94%

Galois actions on regular dessins of small genera

Conder¹,

Jones²,

Streit³

et al. 2013

Rev. Mat. Iberoam.

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“…curves such that the signature of π X is of the form (0; c i , · · · , c r ) where some c i appears exactly an odd number of times. More precisely, we prove the following result, which is a consequence of [4,Theorem 3.1].…”

Section: Introductionmentioning

confidence: 85%

Fields of moduli and fields of definition of odd signature curves

Artebani

Quispe

2012

Arch. Math.

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“…Traduisons brièvement ce qui précède en termes de revêtements de courbes, renvoyant par exemple le lecteur à [3] ou [15] pour des précisions. Avec les notations ci-dessus on commence par définir G = Aut(C), le groupe desK-automorphismes de la courbe (connexe) C, puis la courbe quotient D = C/G.…”

Section: Deux Actions Qui N'en Font Qu'uneunclassified

“…On peut maintenant « oublier » l'isomorphisme G Aut(C) et considérer π comme un « simple » revêtement (« mere cover ») galoisien de groupe G. Le morphisme Gal(K) → Out(G) construit ci-dessus, où K est le corps des modules de C, définit alors l'action (extérieure) du groupe de Galois arithmétique Gal(K) sur le groupe de Galois géométrique Gal(C/D) (voir par exemple [13] ou [15]). …”

Section: Deux Actions Qui N'en Font Qu'uneunclassified

Groupe fondamental des champs algébriques, inertie et action galoisienne

Lochak¹,

Vaquié²

2018

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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On fields of moduli of curves

Cited by 44 publications

References 7 publications

Galois actions on regular dessins of small genera

Galois actions on regular dessins of small genera

Fields of moduli and fields of definition of odd signature curves

Groupe fondamental des champs algébriques, inertie et action galoisienne

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