On computing Belyi maps

Sijsling, Jeroen; Voight, John

doi:10.5802/pmb.5

Cited by 26 publications

(32 citation statements)

References 150 publications

(176 reference statements)

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“…‡ We have three elliptic points. † † An important area where Belyi functions [18,26,27] appear is precisely Shimura curves. Any Belyi covering gives a modular curve with respect to some (not necessarily congruence) subgroup.…”

Section: Heun Function Solutions Of Telescopers Of Rational Functionsmentioning

confidence: 99%

Heun functions and diagonals of rational functions

Abdelaziz¹,

Boukraa²,

Koutschan³

et al. 2020

J. Phys. A: Math. Theor.

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We provide a set of diagonals of simple rational functions of four variables that are seen to be squares of Heun functions. Each time, these Heun functions, obtained by creative telescoping, turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. We even obtained Heun functions that are automorphic forms associated with Shimura curves as solutions of telescopers of rational functions.

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Section: Heun Function Solutions Of Telescopers Of Rational Functionsmentioning

confidence: 99%

Heun functions and diagonals of rational functions

Abdelaziz¹,

Boukraa²,

Koutschan³

et al. 2020

J. Phys. A: Math. Theor.

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“…79) † † One should not confuse these two algebraic curves: the genus-two curve (72) is associated with integrant of the hypergeometric integral (70), when the rational curve (77) is associated with the pullback in the hypergeometric identity (73). ¶ This is a consequence of identity (75).…”

Section: Let Us Consider the Hypergeometric Functionmentioning

confidence: 99%

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

110

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Abstract.We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same 2 F 1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus 2 F 1 hypergeometric function example. We then focus on identities relating the same 2 F 1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale's paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to 3 F 2 , hypergeometric functions, and show that one just reduces to the previous 2 F 1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a 2 F 1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or 2 F 1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

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“…Second, the notion is more conducive to constructive applications: by using branches it becomes straightforward to read off a Weil cocycle from the given rigidification, without having to calculate the canonical model of Aut(X)\X. From the point of view of a computational Esquisse [27], this gain is quite important. Finally, our approach leads us to construct some explicit counterexamples to descent in the case where the marked point P is not smooth, as follows.…”

Section: In His Esquisse D'un Programmementioning

confidence: 99%

On explicit descent of marked curves and maps

Sijsling

Voight

2016

Res. Number Theory

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We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.

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On computing Belyi maps

Cited by 26 publications

References 150 publications

Heun functions and diagonals of rational functions

Heun functions and diagonals of rational functions

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

On explicit descent of marked curves and maps

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