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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2025 Volume 216, Number 4, Pages 113–131 (Mi sm10142)

Finiteness theorems for generalized Jacobians with nontrivial torsion

V. P. Platonovab, V. S. Zhgooncad, G. V. Fedorovea

a Scientific Research Institute for System Analysis of the National Research Centre "Kurchatov Institute", Moscow, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
c Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
d National Research University Higher School of Economics, Moscow, Russia
e Sirius University of Science and Technology, Sochi, Russia

Abstract: Consider a curve $\mathcal C$ defined over an algebraic number field $k$. This work is concerned with the number of generalized Jacobians $J_{\mathfrak{m}}$ of $\mathcal C$ associated with moduli $\mathfrak{m}$ defined over $k$ such that a fixed class of finite order in the Jacobian $J$ of $\mathcal C$ is lifted to a torsion class in the generalized Jacobian $J_{\mathfrak{m}}$. On the one hand it is shown that there are infinitely many generalized Jacobians with the above property, and on the other hand, under some additional constraints on the support of $\mathfrak{m}$ or the structure of $J_{\mathfrak{m}}$, it is shown that the set of generalized Jacobians of this type is finite. In addition, it is proved that there are finitely many generalized Jacobians which have a lift of two given divisors to classes of finite orders in $J_{\mathfrak{m}}$. These results are applied to the problem of the periodicity of continued fractions in the field of formal power series $k((1/x))$ constructed for special elements of the function field $k(\widetilde{\mathcal{C}})$ of a hyperelliptic curve $\widetilde{\mathcal{C}}\colon y^2=f(x)$. In particular, it is shown that for each $n \in \mathbb N$ there is a finite number of monic polynomials $\omega(x) \in k[x]$ of degree at most $n$ such that the element $\omega(x) \sqrt{f(x)}$ has a periodic expansion in a continued fraction.
Bibliography: 14 titles.

Keywords: Jacobian variety, generalized Jacobian, torsion points, continued fractions, hyperelliptic curve.

MSC: Primary 11R58; Secondary 11J70, 11R27

Received: 19.06.2024 and 25.07.2024

DOI: 10.4213/sm10142


 English version:
Sbornik: Mathematics, 2025, 216:4, 560–577


© Steklov Math. Inst. of RAS, 2025