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 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 \(\tilde {\mathcal{C}}:{{y}^{2}} = f(x)\).