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