Geodesic
On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials over number fields
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 32-37 
V. P. Platonov; V. S. Zhgoon; M. M. Petrunin. On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials over number fields. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 32-37. http://geodesic.mathdoc.fr/item/DANMA_2020_493_a6/
@article{DANMA_2020_493_a6,
     author = {V. P. Platonov and V. S. Zhgoon and M. M. Petrunin},
     title = {On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials over number fields},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {32--37},
     year = {2020},
     volume = {493},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_493_a6/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a complete description of fields $\mathbb{K}$ that are quadratic extensions of $\mathbb{Q}$ and of cubic polynomials $f\in\mathbb{K}[x]$ for which a continued fraction expansion of $\sqrt{f}$ in the field of formal power series $\mathbb{K}((x))$ is periodic. We also prove a finiteness theorem for cubic polynomials $f\in\mathbb{K}[x]$ with a periodic expansion of $\sqrt{f}$ over cubic and quartic extensions of $\mathbb{Q}$.

[1] Platonov V.P., “Teoretiko-chislovye svoistva giperellipticheskikh polei i problema krucheniya v yakobianakh giperellipticheskikh krivykh nad polem ratsionalnykh chisel”, UMN, 69:1(415) (2014), 3–38 | DOI | MR | Zbl

[2] Platonov V.P., Petrunin M.M., “Gruppy $S$-edinits i problema periodichnosti nepreryvnykh drobei v giperellipticheskikh polyakh”, Tr. MIAN, 302, 2018, 354–376 | DOI | Zbl

[3] Platonov V.P., Fedorov G.V., “O probleme periodichnosti nepreryvnykh drobei v giperellipticheskikh polyakh”, Matem. sb., 209:4 (2018), 54–94 | DOI | MR | Zbl

[4] Mazur B., “Rational points on modular curves”, Modular Functions of one Variable V, Lecture Notes in Mathematics, eds. J.-P. Serre, Don B. Zagier, Springer, B.–Heidelberg, 107–148 | MR

[5] Kubert Daniel Sion, “Universal bounds on the torsion of elliptic curves”, Proc. London Math. Soc. (3), 33:2 (1976), 193–237 | DOI | MR | Zbl

[6] Kenku Monsur A., “Momose Fumiyuki. Torsion Points on Elliptic Curves Defined over Quadratic Fields”, Nagoya Math. J., 109 (1988), 125–149 | DOI | MR | Zbl

[7] Merel Lo{ïc}, “Bornes pour la torsion des courbes elliptiques sur les corps de nombres”, Invent Math., 124:1 (1996), 437–449 | DOI | MR | Zbl

[8] Mazur B. (with an appendix by Goldfeld D.), “Rational isogenies of prime degree”, Invent Math., 44 (1978), 129–162 | DOI | MR | Zbl

[9] Platonov V.P., Zhgun V.S., Fedorov G.V., “O periodichnosti nepreryvnykh drobei v giperellipticheskikh polyakh nad kvadratichnym polem konstant”, DAN, 482:2 (2018), 137–141 | Zbl

[10] Platonov V.P., Zhgun V.S., Petrunin M.M., Shteinikov Yu.N., “O konechnosti giperellipticheskikh polei so spetsialnymi svoistvami i periodicheskim razlozheniem $\sqrt f $”, DAN, 483:6 (2018), 603–608 | Zbl

[11] Platonov V.P., Petrunin M.M., Shteinikov Yu.N., “O konechnosti chisla ellipticheskikh polei s zadannymi stepenyami $S$-edinits i periodicheskim razlozheniem $\sqrt f $”, DAN, 488:3 (2019), 9–14

[12] Sutherland Andrew, “Constructing Elliptic Curves over Finite Fields with Prescribed Torsion”, Mathematics of Computation, 81:278 (2012), 1131–1147 | DOI | MR | Zbl

[13] Jeon Daeyeol, Kim Chang Heon, Schweizer Andreas, “On the Torsion of Elliptic Curves over Cubic Number Fields”, Acta Arithmetica, 113 (2004), 291–301 | DOI | MR | Zbl

[14] Jeon Daeyeol, Kim Chang Heon, Park Euisung, “On the Torsion of Elliptic Curves over Quartic Number Fields”, J. London Math. Society, 74:1 (2006), 1–12 | DOI | MR | Zbl