A panaroma of the fundamental group of the modular orbifold

Uludağ, A. Muhammed; Zeytin, Ayberk

doi:10.4171/161-1/16

Cited by 7 publications

(12 citation statements)

References 21 publications

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“…is one-to-one; see [12,§10,Satz]. This correspondence is far from being onto as there are many primitive forms of the non-square-free discriminant.…”

Section: Automorphisms Of Narrow Ideal Classesmentioning

confidence: 99%

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Multivariate Lucas polynomials and ideal classes in quadratic number fields

Zeytin¹

2018

Turk J Math

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In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely related to classical Lucas and Fibonacci polynomial sequences and hence to Lucas and Fibonacci numbers. We use one of these families to give a geometric interpretation of the 200 years old class number problems of Gauß, which is equivalent to the study of narrow ideal classes in real quadratic number fields.

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“…is one-to-one; see [12,§10,Satz]. This correspondence is far from being onto as there are many primitive forms of the non-square-free discriminant.…”

Section: Automorphisms Of Narrow Ideal Classesmentioning

confidence: 99%

“…Let us remark that the covering category consisting of étale covers of the modular orbifold is so rich that the whole absolute Galois group can be recovered from it; see [10]. In [9], a project that outlines a is discussed.…”

Section: çArksmentioning

confidence: 99%

Multivariate Lucas polynomials and ideal classes in quadratic number fields

Zeytin¹

2018

Turk J Math

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“…This association of a dessin to a triangulation is very natural. For more details the reader is referred to Chapter 15 of the present volume [87].…”

Section: Some Non-linear Actions Of the Absolute Galois Groupsmentioning

confidence: 99%

“…with finitely generated fundamental group) which are of arithmetic interest. See also Chapter 15 of the present volume [87]. moduli space M 0,4 of the sphere with four punctures.…”

Section: Introductionmentioning

confidence: 98%

Actions of the absolute Galois group

A'Campo¹,

Ji²,

Papadopoulos³

2016

IRMA Lectures in Mathematics and Theoretical Physics

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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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“…Our aim in this paper is to clarify the connections between these four actions. See [23] or [24] for an overview of the related subjects from a wider perspective. In particular, the actions in consideration will play a crucial role in observing non-trivial relations between Teichmüller theory and arithmetic.…”

Section: Introductionmentioning

confidence: 99%