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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 1(415), Pages 3–38 (Mi umn9563)  

This article is cited in 20 scientific papers (total in 22 papers)

Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field

V. P. Platonovab

a Steklov Mathematical Institute of Russian Academy of Sciences
b Scientific Research Institute for System Studies of Russian Academy of Sciences

Abstract: In the past four years a theory has been developed for finding fundamental units in hyperelliptic fields, and on basis of this theory innovative and efficient algorithms for computing them have been constructed and implemented. A new local-global principle was discovered which gives a criterion for the existence of non-trivial units in hyperelliptic fields. The natural connection between the problem of computing fundamental units and the problem of torsion in Jacobian varieties of hyperelliptic curves over the rational number field has led to breakthrough results in the solution of this problem. The main results in the present survey were largely obtained using a symbiosis of deep theory, efficient algorithms, and supercomputing. Such a symbiosis will play an ever increasing role in the mathematics of the 21st century.
Bibliography: 27 titles.

Keywords: fundamental units, hyperelliptic fields, local-global principle, Jacobian varieties, hyperelliptic curves, torsion problem in Jacobians, fast algorithms, continued fractions.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00190
13-01-12402
This work was supported by the Russian Foundation for Basic Research (grant nos. 12-01-0190 and 13-01-12402).


DOI: https://doi.org/10.4213/rm9563

Full text: PDF file (684 kB)
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English version:
Russian Mathematical Surveys, 2014, 69:1, 1–34

Bibliographic databases:

Document Type: Article
UDC: 511.6+512.74
MSC: 11G30, 11R27, 14H40
Received: 15.11.2013

Citation: V. P. Platonov, “Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field”, Uspekhi Mat. Nauk, 69:1(415) (2014), 3–38; Russian Math. Surveys, 69:1 (2014), 1–34

Citation in format AMSBIB
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    2. V. P. Platonov, M. M. Petrunin, “New curves of genus 2 over the field of rational numbers whose Jacobians contain torsion points of high order”, Dokl. Math., 91:2 (2015), 220–221  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. S. I. Adyan, V. V. Benyash-Krivets, V. M. Bukhshtaber, E. I. Zelmanov, V. V. Kozlov, G. A. Margulis, S. P. Novikov, A. N. Parshin, G. Prasad, A. S. Rapinchuk, L. D. Faddeev, V. I. Chernousov, “Vladimir Petrovich Platonov (k 75-letiyu so dnya rozhdeniya)”, Chebyshevskii sb., 16:4 (2015), 6–10  mathnet
    4. M. M. Petrunin, “Vychislenie fundamentalnykh $S$-edinits v giperellipticheskikh polyakh roda $2$ i problema krucheniya v yakobianakh giperellipticheskikh krivykh”, Chebyshevskii sb., 16:4 (2015), 250–283  mathnet  elib
    5. V. P. Platonov, M. M. Petrunin, “Fundamental $S$-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves”, Dokl. Math., 92:3 (2015), 667–669  mathnet  crossref  crossref  mathscinet  zmath  isi  isi  elib  scopus
    6. V. P. Platonov, G. V. Fedorov, “S-Units and Periodicity of Continued Fractions in Hyperelliptic Fields”, Dokl. Math., 92:3 (2015), 752–756  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    7. Proc. Steklov Inst. Math., 292 (2016), 63–93  mathnet  crossref  crossref  mathscinet  isi  elib
    8. V. P. Platonov, M. M. Petrunin, “$S$-Units and periodicity in quadratic function fields”, Russian Math. Surveys, 71:5 (2016), 973–975  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. G. V. Fedorov, “On the periodicity of continued fractions in hyperelliptic fields”, Advances in dynamical systems and control, Stud. Syst. Decis. Control, 69, Springer, Cham, 2016, 141–157  crossref  mathscinet  zmath  isi  scopus
    10. V. P. Platonov, “On new properties of Hankel matrices over the field of rational numbers”, Russian Math. Surveys, 72:5 (2017), 963–964  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. M. M. Petrunin, “$S$-units and the periodicity of the square root in hyperelliptic fields”, Dokl. Math., 95:3 (2017), 222–225  crossref  mathscinet  zmath  isi  scopus
    12. V. P. Platonov, G. V. Fedorov, “On the periodicity of continued fractions in hyperelliptic fields”, Dokl. Math., 95:3 (2017), 254–258  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    13. V. S. Zhgun, “Obobschennye yakobiany i nepreryvnye drobi v giperellipticheskikh polyakh”, Chebyshevskii sb., 18:4 (2017), 209–221  mathnet  crossref
    14. Yu. V. Kuznetsov, Yu. N. Shteinikov, “O nekotorykh svoistvakh nepreryvnykh periodicheskikh drobei s nebolshoi dlinoi perioda, svyazannykh s giperellipticheskimi polyami i $S$-edinitsami”, Chebyshevskii sb., 18:4 (2017), 261–268  mathnet  crossref
    15. K. Daowsud, T. A. Schmidt, “Continued fractions for rational torsion”, J. Number Theory, 189 (2018), 115–130  crossref  mathscinet  zmath  isi  scopus
    16. V. P. Platonov, G. V. Fedorov, “On the problem of periodicity of continued fractions in hyperelliptic fields”, Sb. Math., 209:4 (2018), 519–559  mathnet  crossref  crossref  adsnasa  isi  elib
    17. Platonov V.P. Petrunin M.M., “On New Arithmetic Properties of Determinants of Hankel Matrices”, Dokl. Math., 98:1 (2018), 370–372  crossref  zmath  isi  scopus
    18. V. P. Platonov, M. M. Petrunin, “Groups of $S$-units and the problem of periodicity of continued fractions in hyperelliptic fields”, Proc. Steklov Inst. Math., 302 (2018), 336–357  mathnet  crossref  crossref  isi  elib
    19. Platonov V.P., Zhgoon V.S., Fedorov G.V., “On the Periodicity of Continued Fractions in Hyperelliptic Fields Over Quadratic Constant Field”, Dokl. Math., 98:2 (2018), 430–434  crossref  mathscinet  zmath  isi
    20. Platonov V.P., Fedorov G.V., “An Infinite Family of Curves of Genus 2 Over the Field of Rational Numbers Whose Jacobian Varieties Contain Rational Points of Order 28”, Dokl. Math., 98:2 (2018), 468–471  crossref  zmath  isi
    21. Platonov V.P. Zhgoon V.S. Petrunin M.M. Shteinikov Yu.N., “On the Finiteness of Hyperelliptic Fields With Special Properties and Periodic Expansion of F”, Dokl. Math., 98:3 (2018), 641–645  crossref  zmath  isi
    22. Zannier U., “Hyperelliptic Continued Fractions and Generalized Jacobians”, Am. J. Math., 141:1 (2019), 1–40  mathscinet  zmath  isi
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