Dessins d’enfants and origami curves

Herrlich, Frank; Schmithüsen, Gabriela

doi:10.4171/055-1/19

Cited by 17 publications

(22 citation statements)

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“…Examples of dessins d'enfants associated to some specific Belyi morphisms are contained in [43]. We also refer the reader to the survey [40] in volume I of this Handbook and to the surveys by L. Schneps, in particular [76].…”

Section: The Action Of the Absolute Galois Group On Dessins D'enfantsmentioning

confidence: 99%

Actions of the absolute Galois group

A'Campo¹,

Ji²,

Papadopoulos³

2016

Handbook of Teichmüller Theory, Volume VI

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We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichmüller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his Récoltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichmüller tower, and other related objects, he considered the actions of Γ Q on them or on theirétale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automorphism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of Γ Q , in particular in homotopy theory.

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Section: The Action Of the Absolute Galois Group On Dessins D'enfantsmentioning

confidence: 99%

Actions of the absolute Galois group

A'Campo¹,

Ji²,

Papadopoulos³

2016

Handbook of Teichmüller Theory, Volume VI

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“…-One generalization of Belyȋ maps is given by covers called origamis: covers of elliptic curves that are unramified away from the origin. For a more complete account on origamis, see Herrlich-Schmithüsen [74]; Belyȋ maps can be obtained from origamis by a degeneration process [74, §8].…”

Section: Further Topics and Generalizationsmentioning

confidence: 99%

On computing Belyi maps

Sijsling¹,

Voight²

2015

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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We survey methods to compute three-point branched covers of the projective line, also known as Belyȋ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtements de la droite projective ramifiés sur au plus trois points, aussi appelés les morphismes de Belyȋ. Ces méthodes comprennent une approche directe, qui revientà la solution d'un système d'équations polynomiales, ainsi que des méthodes analytiques complexes, méthodes de formes modulaires, et méthodes p-adiques. En chemin, nous posons quelques questions et donnons de nombreux exemples.

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“…A branched cover of an elliptic curve branched at one point is known as an origami. Origamis can be represented pictorially as in Figure 1, and the topological viewpoint has been much studied, for example by Herrlich and Schmithüsen in [HS09]. Conversely, if E is the elliptic curve C/Z[i], a square tile arrangement like the one shown in Figure 1 can be interpreted as a smooth curve C (say of genus g) with a differential ω and a map f : C → E satisfying f * dx y = ω; hence the square tiling determines a flat structure on a curve of genus g ≥ 1, with finitely many cone points.…”

Section: Introductionmentioning

confidence: 99%

Period computations for covers of elliptic curves

Rubinstein-Salzedo

2014

Math. Comp.

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In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and using these relations to approximate the coefficients to high precision. We then conjecture which algebraic numbers the coefficients are, and then we prove this conjecture to be correct.

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Dessins d’enfants and origami curves

Cited by 17 publications

References 0 publications

Actions of the absolute Galois group

Actions of the absolute Galois group

On computing Belyi maps

Period computations for covers of elliptic curves

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