ON THE FINITENESS OF THE SET OF GENERALIZED JACOBIANS WITH NONTRIVIAL TORSION POINTS OVER ALGEBRAIC NUMBER FIELDS
- Authors: Platonov V.P.1,2, Fedorov G.V.1,3,4, Zhgoon V.S.1,5
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Affiliations:
- Federal State Institution Scientific Research Institute for System Analysis of the Russian Academy of Sciences
- Steklov Mathematical Institute Russian Academy of Sciences
- National Research University Higher School of Economics
- Moscow Institute of Physics and Technology (National Research University)
- Lomonosov Moscow State University
- Issue: Vol 513, No 1 (2023)
- Pages: 66-70
- Section: MATHEMATICS
- URL: https://rjeid.com/2686-9543/article/view/647897
- DOI: https://doi.org/10.31857/S2686954323700285
- EDN: https://elibrary.ru/CLLXDV
- ID: 647897
Cite item
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
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