Binary quadratic forms as dessins
Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 445-469.

Nous montrons que la classe de toute forme quadratique binaire indéterminée et primitive est représentée de façon naturelle par un graphe infini (appellé çark) avec un unique cycle, plongé dans une couronne conforme. Ce cycle est appelé le rachis du çark. Le choix d’un arc d’un çark donné spécifie une forme quadratique binaire indéterminée dans la classe représentée par le çark. Les formes réduites dans la classe représentée par un çark correspondent à certains arcs distingués sur son rachis. La réduction de Gauss est le processus de déplacement de l’arc vers la direction du rachis du çark. Les classes ambiguës et réciproques sont représentées par des çarks ayant une symétrie. Les çarks périodiques représentent les classes des formes non-primitives.

We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every choice of an edge of a fixed çark specifies an indefinite binary quadratic form in the class represented by the çark. Reduced forms in the class represented by a çark correspond to some distinguished edges on its spine. Gauss reduction is the process of moving the edge in the direction of the spine of the çark. Ambiguous and reciprocal classes are represented by çarks with symmetries. Periodic çarks represent classes of non-primitive forms.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.987
Classification : 11H55,  05C10
Mots clés : binary quadratic forms, dessins d’enfants, bipartite ribbon graphs, çarks, ambiguous forms, reciprocal forms, Markoff number
@article{JTNB_2017__29_2_445_0,
     author = {A. Muhammed Uluda\u{g} and Ayberk Zeytin and Merve Durmu\c{s}},
     title = {Binary quadratic forms as dessins},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {445--469},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {2},
     year = {2017},
     doi = {10.5802/jtnb.987},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.987/}
}
A. Muhammed Uludağ; Ayberk Zeytin; Merve Durmuş. Binary quadratic forms as dessins. Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 445-469. doi : 10.5802/jtnb.987. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.987/

[1] G. V. Belyĭ On Galois extensions of a maximal cyclotomic field, Math. USSR, Izv., Volume 14 (1980), pp. 247-256 | Article

[2] Seifeddine Bouallegue; Mongi Naimi On primitive words, Int. J. Algebra, Volume 4 (2010) no. 13-16, pp. 693-707

[3] Johannes Buchmann; Ulrich Vollmer Binary quadratic forms. An algorithmic approach., Algorithms and Computation in Mathematics, Volume 20, Springer, 2007, xiv+318 pages

[4] Henri Cohen A course in computational algebraic number theory, Graduate Texts in Mathematics, Volume 138, Springer, 1993, xxi+534 pages

[5] John Horton Conway The sensual (quadratic) form, The Carus Mathematical Monographs, Volume 26, The Mathematical Association of America, 1997, xiii+152 pages

[6] Merve Durmuş Farey graph and binary quadratic forms (2012) (Ph. D. Thesis)

[7] Jean-Pierre Duval Génération d’une section des classes de conjugaison et arbre des mots de Lyndon de longueur bornée, Theor. Comput. Sci., Volume 60 (1988) no. 3, pp. 255-283 | Article

[8] Harold Fredricksen; Irving Kessler Lexicographic compositions and de Bruijn sequences, J. Comb. Theory, Volume 22 (1977), pp. 17-30 | Article

[9] Carl Friedrich Gauss Disquisitiones arithmeticae, Yale University Press, 1966, xx+472 pages

[10] Team Infomod Sunburst version 0 (2013) (available at http://math.gsu.edu.tr/azeytin/infomod/node/3)

[11] Svetlana Katok; Ilie Ugarcovici Symbolic dynamics for the modular surface and beyond, Bull. Am. Math. Soc., Volume 44 (2007) no. 1, pp. 87-132 | Article

[12] Felix Klein Über die Transformation elfter Ordnung der elliptischen Functionen, Clebsch Ann., Volume XV (1879), pp. 533-555

[13] Ravi S. Kulkarni An arithmetic-geometric method in the study of the subgroups of the modular group, Am. J. Math., Volume 113 (1991) no. 6, pp. 1053-1133 | Article

[14] Sergei K. Lando; Alexander K. Zvonkin Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, Volume 141, Springer, 2004, xv+455 pages

[15] Percy Alexander MacMahon Applications of a Theory of Permutations in Circular Procession to the Theory of Numbers, Proc. Lond. Math. Soc., Volume s1-23 (1891) no. 1, pp. 305-318 | Article

[16] M.Aslam Malik; M.Asim Zafar Real quadratic irrational numbers and modular group action, Southeast Asian Bull. Math., Volume 35 (2011) no. 3, pp. 439-445

[17] Yuri Ivanovich Manin Real multiplication and noncommutative geometry (ein Alterstraum), The legacy of Niels Henrik Abel, Springer, 2004, pp. 685-727

[18] Qaiser Mushtaq Modular group acting on real quadratic fields, Bull. Aust. Math. Soc., Volume 37 (1988) no. 2, pp. 303-309 | Article

[19] Peter Sarnak Reciprocal geodesics, Analytic number theory. A tribute to Gauss and Dirichlet (Clay Mathematics Proceedings) Volume 7, American Mathematical Society, 2007, pp. 217-237

[20] Joe Sawada Generating Bracelets in Constant Amortized Time, SIAM J. Comput., Volume 31 (2001) no. 1, pp. 259-268 | Article

[21] Neil James Alexander Sloane The On-Line Encyclopedia of Integer Sequences (published electronically at http://oeis.org/)

[22] The PARI Group PARI/GP version 2.5.0, 2012 (available at http://pari.math.u-bordeaux.fr/)

[23] A. Muhammed Uludağ The modular group and its actions, Volume I, Handbook of group actions (Advanced Lectures in Mathematics) Volume 31, International Press and Higher Education Press, 2015, pp. 333-370

[24] A. Muhammed Uludağ; Ayberk Zeytin A panorama of the fundamental group of the modular orbifold, Handbook of Teichmüller Theory, Vol. VI (IRMA Lectures in Mathematics and Theoretical Physics) Volume 27, European Mathematical Society, 2016, pp. 501-519

[25] Don B. Zagier Zetafunktionen und quadratische Körper. Eine Einführung in die höhere Zahlentheorie, Hochschultext, Springer, 1981, ix+144 pages

[26] Don B. Zagier New points of view on the selberg zeta function, Proceedings of Japanese-German Seminar (2002), pp. 1-10

[27] Ayberk Zeytin On reduction theory of binary quadratic forms, Publ. Math., Volume 89 (2016), pp. 203-221

[28] Ayberk Zeytin; Hakan Ayral; A. Muhammed Uludağ InfoMod: A visual and computational approach to Gauss’ binary quadratic forms (2017) (https://arxiv.org/abs/1704.00902)

Cité par document(s). Sources :