ON THE FINITENESS OF THE SET OF GENERALIZED JACOBIANS WITH NONTRIVIAL TORSION POINTS OVER ALGEBRAIC NUMBER FIELDS

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Abstract

For a smooth projective curve \(\mathcal{C}\) defined over algebraic number field k, we investigate the question of finiteness of the set of generalized Jacobians \({{J}_{\mathfrak{m}}}\) of a curve \(\mathcal{C}\) associated with modules \(\mathfrak{m}\) defined over k such that a fixed divisor representing a class of finite order in the Jacobian J of the curve \(\mathcal{C}\) provides the torsion class in the generalized Jacobian \({{J}_{\mathfrak{m}}}\). Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of \(\mathfrak{m}\), as well as on the conditions on the field \(k\). These results were applied to the problem of the periodicity of a continuous fraction decomposition constructed in the field of formal power series \(k((1{\text{/}}x))\), for the special elements of the field of functions \(k(\tilde {\mathcal{C}})\) of the hyperelliptic curve \(\tilde {\mathcal{C}}:{{y}^{2}} = f(x)\).

About the authors

V. P. Platonov

Federal State Institution Scientific Research Institute for System Analysis of the Russian Academy of Sciences; Steklov Mathematical Institute Russian Academy of Sciences

Author for correspondence.
Email: platonov@mi-ras.ru
Russian Federation, Moscow; Russian Federation, Moscow

G. V. Fedorov

Federal State Institution Scientific Research Institute for System Analysis of the Russian Academy of Sciences; National Research University Higher School of Economics; Moscow Institute of Physics and Technology (National Research University)

Author for correspondence.
Email: zhgoon@mail.ru
Russian Federation, Moscow; Russian Federation, Moscow; Russian Federation, Moscow

V. S. Zhgoon

Federal State Institution Scientific Research Institute for System Analysis of the Russian Academy of Sciences; Lomonosov Moscow State University

Author for correspondence.
Email: fedorov@mech.math.msu.su
Russian Federation, Moscow; Russian Federation, Moscow

References

  1. Платонов В.П. Теоретико-числовые свойства гиперэллиптических полей и проблема кручения в якобианах гиперэллиптических кривых над полем рациональных чисел // УМН. 2014. V. 69:1 (415). P. 3–38.
  2. Платонов В.П., Федоров Г.В. О проблеме классификации многочленов f с периодическим разложением в непрерывную дробь в гиперэллиптических полях // Известия Российской академии наук. Серия математическая. 2021. Т. 85. № 5. С. 152–189.
  3. Платонов В.П., Федоров Г.В. О проблеме периодичности непрерывных дробей в гиперэллиптических полях // Матем. сб. 2018. Т. 209. № 4. С. 54–94.
  4. Schmidt W.M. On continued fractions and diophantine approximation in power series fields // Acta arithmetica.2000. V. 95:2. P. 139–166.
  5. Rosenlicht M. Generalized jacobian varieties // Annals of Mathematics. 1954. P. 505–530.
  6. Zannier U. Hyperelliptic continued fractions and generalized Jacobians // American Journal of Mathematics. 2019. V. 141:1. P. 1–40.
  7. Серр Ж.П. Алгебраические группы и поля классов. М.: Мир, 1968. 278 с.
  8. Ленг С. Алгебраические числа. М.: Мир, 1966. 226 с.

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