Continued Fractions and the Classification Problem for Elliptic Fields Over Quadratic Fields of Constants

G. V. Fedorov

Matematičeskie zametki, Tome 114 (2023) no. 6, pp. 873-893.

Résumé

The theory of periodicity of functional continued fractions has found deep applications to the problem of finding and constructing fundamental units and $S$-units, the problem of describing points of finite order on elliptic curves, and the torsion problem in Jacobians of hyperelliptic curves. Functional continued fractions are also of interest from the point of view of arithmetic applications, in particular, to solving norm equations or Pell-type functional equations. In this paper, given any quadratic number field $K$, all square-free fourth-degree polynomials $f(x) \in K[x]$ are described such that $\sqrt{f}$ has periodic continued fraction expansion in the field $K((x))$ of formal power series and the elliptic field $L=K(x)(\sqrt{f})$ has a fundamental $S$-unit of degree $m$, $2 \le m \le 12$, $m \ne 11$, where the set $S$ consists of two conjugate valuations defined on $L$ and related to the uniformizing element $x$ of the field $K(x)$.

Détail

Keywords: continued fraction, hyperelliptic curve, fundamental unit, modular curve, divisor class group, torsion subgroup in Jacobian.

Bibliographie

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

[3] V. V. Benyash-Krivets, V. P. Platonov, “Gruppy $S$-edinits v giperellipticheskikh polyakh i nepreryvnye drobi”, Matem. sb., 200:11 (2009), 15–44 | DOI | MR | Zbl

[4] V. P. Platonov, M. M. Petrunin, “$S$-edinitsy i periodichnost v kvadratichnykh funktsionalnykh polyakh”, UMN, 71:5 (431) (2016), 181–182 | DOI | MR | Zbl

[5] V. P. Platonov, G. V. Fedorov, “O periodichnosti nepreryvnykh drobei v ellipticheskikh polyakh”, Dokl. AN, 475:2 (2017), 133–136 | DOI | MR

[6] W. W. Adams, M. J. Razar, “Multiples of points on elliptic curves and continued fractions”, Proc. London Math. Soc. (3), 41:3 (1980), 481–498 | DOI | MR

[7] A. J. van der Poorten, X. C. Tran, “Periodic continued fractions in elliptic function fields”, Algorithmic Number Theory (Sydney, 2002), Lecture Notes in Comput. Sci., 2369, Springer, Berlin, Heidelberg, 2002, 390–404 | DOI | MR

[8] A. J. van der Poorten, “Periodic continued fractions and elliptic curves”, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004, 353–365 | MR

[9] U. Zannier, “Hyperelliptic continued fractions and generalized Jacobians”, Amer. J. Math., 141:1 (2019), 1–40 | DOI | MR

[10] A. N. W. Hone, “Continued fractions and Hankel determinants from hyperelliptic curves”, Comm. Pure Appl. Math., 74:11 (2021), 2310–2347 | DOI | MR

[11] T. G. Berry, “On periodicity of continued fractions in hyperelliptic function fields”, Arch. Math. (Basel), 55:3 (1990), 259–266 | DOI | MR

[12] W. M. Schmidt, “On continued fractions and Diophantine approximation in power series fields”, Acta Arith., 95:2 (2000), 139–166 | DOI | MR

[13] V. P. Platonov, G. V. Fedorov, “O probleme klassifikatsii mnogochlenov $f$ s periodicheskim razlozheniem $\sqrt{f}$ v nepreryvnuyu drob v giperellipticheskikh polyakh”, Izv. RAN. Ser. matem., 85:5 (2021), 152–189 | DOI | MR

[14] M. M. Petrunin, “$S$-edinitsy i periodichnost kvadratnogo kornya v giperellipticheskikh polyakh”, Dokl. AN, 474:2 (2017), 155–158 | MR

[15] V. P. Platonov, G. V. Fedorov, “Kriterii periodichnosti nepreryvnykh drobei klyuchevykh elementov giperellipticheskikh polei”, Chebyshevskii sb., 20:1 (2019), 248–260 | DOI

[16] G. V. Fedorov, “O probleme opisaniya elementov ellipticheskikh polei s periodicheskim razlozheniem v nepreryvnuyu drob nad kvadratichnymi polyami konstant”, Dokl. RAN. Matem., inform., prots. upr., 505 (2022), 56–62 | DOI

[17] V. P. Platonov, V. S. Zhgun, M. M. Petrunin, Yu. N. Shteinikov, “O konechnosti giperellipticheskikh polei so spetsialnymi svoistvami i periodicheskim razlozheniem $\sqrt{f}$”, Dokl. AN, 483:6 (2018), 609–613 | DOI

[18] V. P. Platonov, M. M. Petrunin, Yu. N. Shteinikov, “O konechnosti chisla ellipticheskikh polei s zadannymi stepenyami $S$-edinits i periodicheskim razlozheniem $\sqrt{f}$”, Dokl. AN, 488:3 (2019), 237–242 | DOI | MR

[19] V. P. Platonov, M. M. Petrunin, “O konechnosti chisla periodicheskikh razlozhenii v nepreryvnuyu drob $\sqrt f$ dlya kubicheskikh mnogochlenov nad polyami algebraicheskikh chisel”, Dokl. RAN. Matem., inform., prots. upr., 495 (2020), 48–54 | DOI | Zbl

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

[21] B. Mazur, “Rational points on modular curves”, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., 601, Springer-Verlag, Berlin–New York, 1977, 107–148 | MR

[22] M. A. Kenku, F. Momose, “Torsion points on elliptic curves defined over quadratic fields”, Nagoya Math. J., 109 (1988), 125–149 | DOI | MR

[23] G. V. Fedorov, “Ob otsenkakh dlin periodov funktsionalnykh nepreryvnykh drobei nad algebraicheskimi chislovymi polyami”, Chebyshevckii sb., 25:2 (2023) (to appear)

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

[25] Z. L. Scherr, Rational Polynomial Pell Equations, Thesis, The University of Michigan, 2013

[26] G. V. Fedorov, “O dline perioda funktsionalnoi nepreryvnoi drobi nad chislovym polem”, Dokl. RAN. Matem., inform., prots. upr., 495 (2020), 78–81 | DOI | Zbl

[27] A. Meurer et al., “SymPy: symbolic computing in Python”, PeerJ Computer Science, 3 (2017), e103 | DOI

[28] SymPy 1.11 Documentation, https://docs.sympy.org/latest/index.html