On the finiteness of the set of generalized Jacobians with nontrivial torsion points over algebraic number fields

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 $J_m$ of a curve $\mathcal{C}$ associated with modules $m$ 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 $J_m$. 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 $m$, 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 $k((1/x))$, for the special elements of the field of functions $k(\tilde{\mathcal{C}})$ of the hyperelliptic curve $\tilde{\mathcal{C}}:y^2=f(x)$.