On the Finiteness of the Set of Generalized Jacobians with Nontrivial Torsion Points over Algebraic Number Fields

Abstract

For a smooth projective curve $\mathcal{C}$ defined over an algebraic number field k, we investigate the finiteness of the set of generalized Jacobians ${{J}_{\mathfrak{m}}}$ of $\mathcal{C}$ associated with modules $\mathfrak{m}$ defined over $k$ such that a fixed divisor representing a class of finite order in the Jacobian J of $\mathcal{C}$ 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 are applied to the problem of the periodicity of a continued fraction expansion constructed in the field of formal power series $k((1{\text{/}}x))$ for special elements of the field of functions $k(\tilde {\mathcal{C}})$ of the hyperelliptic curve $\tilde {\mathcal{C}}:{{y}^{2}} = f(x)$.