Binary quadratic forms as dessins

Uludağ, A. Muhammed; Zeytin, Ayberk; Durmuş, Merve

doi:10.5802/jtnb.987

Cited by 9 publications

(17 citation statements)

References 16 publications

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“…Recently, Uludağ, Zeytin, and Durmuş (UZD) gave another construction of binary necklaces attached to classes of indefinite forms [13] . With each class, they associate a graph called a çark.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory

Smith

2018

Journal of Number Theory

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Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure of Gauss's reduction theory into that of Zagier's. From this perspective, Zagier-reduction becomes a refinement of Gauss-reduction.

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Section: Summary Of Resultsmentioning

confidence: 99%

Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory

Smith

2018

Journal of Number Theory

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“…From the one-to-one correspondence between dessins d'enfants and covers of the Riemann sphere S 2 with ramification points at {0, 1, ∞}, we get an action of Γ Q on the algebraic fundamental group of S 2 − {0, 1, ∞}. But this fundamental group is the first level of the Teichmüller tower, since it is the fundamental group of the 3 In the paper [88], it is shown that infinite coverings, although no more algebraic, do convey essential arithmetic information. What is studied in that paper is a version of the "annular uniformization" of the thrice-punctured sphere that is developed in Strebel's book on quadratic differentials [78].…”

Section: Introductionmentioning

confidence: 99%

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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“…Now by its very definition, this tree admits a PSL(2, Z)-action, and a quotient graph by a subgroup of finite or infinite index G < PSL(2, Z) has been named by us a modular graph 1 [24]. Thus the modular graph G\F sits inside the curve G\H in a standard way.…”

Section: Its Quotient Space Is Called Thementioning

confidence: 99%

“…We firmly believe that such an interpretation does exist and is fruitful, however, we don't know how to make this. Instead we will give an alternative arithmetic interpretation of hyperbolic Z-covers from [24].…”

Section: Arithmetic Of Non-tame Subgroupsmentioning

confidence: 99%