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

This article is cited in 33 scientific papers (total in 35 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:

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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\jour Uspekhi Mat. Nauk
\yr 2014
\vol 69
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\pages 3--38
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. I. Adian, V. V. Benyash-Krivets, V. M. Buchstaber, 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 (on his 75th birthday)”, Russian Math. Surveys, 70:1 (2015), 197–201  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathscinet  adsnasa  isi  elib
    17. V. P. Platonov, M. M. Petrunin, “On new arithmetic properties of determinants of Hankel matrices”, Dokl. Math., 98:1 (2018), 370–372  mathnet  crossref  crossref  zmath  isi  elib  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  mathscinet  isi  elib
    19. V. P. Platonov, V. S. Zhgoon, G. V. Fedorov, “On the periodicity of continued fractions in hyperelliptic fields over quadratic constant field”, Dokl. Math., 98:2 (2018), 430–434  crossref  zmath  isi
    20. V. P. Platonov, G. V. Fedorov, “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. V. P. Platonov, V. S. Zhgoon, M. M. Petrunin, Yu. N. Shteinikov, “On the finiteness of hyperelliptic fields with special properties and periodic expansion of $\sqrt f$”, Dokl. Math., 98:3 (2018), 641–645  mathnet  crossref  crossref  zmath  isi  elib
    22. G. V. Fedorov, “Periodicheskie nepreryvnye drobi i $S$-edinitsy s normirovaniyami vtoroi stepeni v giperellipticheskikh polyakh”, Chebyshevskii sb., 19:3 (2018), 282–297  mathnet  crossref  elib
    23. U. Zannier, “Hyperelliptic continued fractions and generalized Jacobians”, Am. J. Math., 141:1 (2019), 1–40  mathscinet  zmath  isi
    24. V. P. Platonov, M. M. Petrunin, “On infinite-dimensional integer Hankel matrices”, Dokl. Math., 99:2 (2019), 218–220  crossref  isi
    25. V. P. Platonov, G. V. Fedorov, “On s-units for linear valuations and the periodicity of continued fractions of generalized type in hyperelliptic fields”, Dokl. Math., 99:3 (2019), 277–281  crossref  isi
    26. V. P. Platonov, M. M. Petrunin, Yu. N. Shteinikov, “On the finiteness of the number of elliptic fields with given degrees of s-units and periodic expansion of root F”, Dokl. Math., 100:2 (2019), 440–444  crossref  isi
    27. V. P. Platonov, G. V. Fedorov, “Kriterii periodichnosti nepreryvnykh drobei klyuchevykh elementov giperellipticheskikh polei”, Chebyshevskii sb., 20:1 (2019), 248–260  mathnet  crossref
    28. G. V. Fedorov, “Ob ogranichennosti dlin periodov nepreryvnykh drobei klyuchevykh elementov giperellipticheskikh polei nad polem ratsionalnykh chisel”, Chebyshevskii sb., 20:4 (2019), 357–370  mathnet  crossref
    29. G. V. Fedorov, “On $S$-units for valuations of the second degree in hyperelliptic fields”, Izv. Math., 84:2 (2020), 392–435  mathnet  crossref  crossref  mathscinet  isi  elib
    30. V. P. Platonov, G. V. Fedorov, “On the problem of classification of periodic continued fractions in hyperelliptic fields”, Russian Math. Surveys, 75:4 (2020), 785–787  mathnet  crossref  crossref  mathscinet  isi  elib
    31. T. Ayano, V. M. Buchstaber, “Analytical and number-theoretical properties of the two-dimensional sigma function”, Chebyshevskii sb., 21:1 (2020), 9–50  mathnet  crossref  mathscinet
    32. V. P. Platonov, M. M. Petrunin, Yu. N. Shteinikov, “Periodicheskie elementy $\sqrt{f}$ v ellipticheskikh polyakh s polem konstant nulevoi kharakteristiki”, Chebyshevskii sb., 21:1 (2020), 273–296  mathnet  crossref  mathscinet
    33. G. V. Fedorov, “O semeistvakh giperellipticheskikh krivykh nad polem ratsionalnykh chisel, yakobiany kotorykh soderzhat tochki krucheniya dannykh poryadkov”, Chebyshevskii sb., 21:1 (2020), 322–340  mathnet  crossref
    34. Platonov V.P., Petrunin M.M., Zhgoon V.S., “On the Problem of Periodicity of Continued Fraction Expansions of Root F For Cubic Polynomials Over Number Fields”, Dokl. Math., 102:1 (2020), 288–292  crossref  isi
    35. V. P. Platonov, G. V. Fedorov, “On the classification problem for polynomials $f$ with a periodic continued fraction expansion of $\sqrt{f}$ in hyperelliptic fields”, Izv. Math., 85:5 (2021), 972–1007  mathnet  crossref  crossref  isi
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