Explicit Algebraic Coverings of a Pointed Torus

Anema, Ane S.I.; Top, Jaap

doi:10.1007/978-1-4614-6403-7_5

Cited by 1 publication

(4 citation statements)

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“…with marked point (0, 0). The equation (3.3) is unique up to scaling the coefficients by u = 0 according to (p 2 , p 3 , p 4 ) → (u 2 a 2 , u 3 a 3 , u 4 a 4 ), showing that the moduli spaces M 1,2 of genus 1 curves with two marked points is isomorphic to the weighted projective space P (2,3,4).…”

Section: Complex Analytic Methodsmentioning

confidence: 99%

“…We find several solutions in F 49 but only one solution lifts p-adically without additional effort; it turns out the Jacobian of the corresponding system of equations is not of full rank. After some effort (see also Section 8), we recognize this cover as an M 11 -cover with ramification (3,5,6), defined over the number field Q(α) where α 7 − α 6 − 8α 5 + 21α 4 + 6α 3 − 90α 2 + 60α + 60 = 0.…”

Section: P-adic Methodsmentioning

confidence: 99%

“…Klein [90] and Fricke-Klein [56] calculated some explicit Belyȋ maps, most notably the icosahedral Galois Belyȋ map P 1 → P 1 of degree 60 [90, I, 2, §13-14]. These appeared when constructing what we would today call modular functions associated with the triangle groups ∆ (2,3,5) and ∆(2, 4, 5) (see Section 4). This in turn allowed them to find a solution to the quintic equation by using analytic functions.…”

Section: Background and Applicationsmentioning

confidence: 99%

“…An exciting family of special origamis was considered by Anema-Top [3]: they consider the elliptic curve E : y 2 = x 3 + ax + b over the scheme B : 4a 3 + 27b 2 = 1 defined by the constant non-vanishing discriminant 1 of E. Considering the torsion subschemes E[n] over B, one obtains a family of covers over the base elliptic curve B of j-invariant 0 that is only ramified above the point at infinity and whose Galois groups are subgroups of special linear groups. It would be very interesting to deform this family to treat the case of arbitrary base curves, though it is not clear how to achieve this.…”

Section: Further Topics and Generalizationsmentioning

confidence: 99%

See 3 more Smart Citations

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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Section: Complex Analytic Methodsmentioning

confidence: 99%

Section: P-adic Methodsmentioning

confidence: 99%