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

For a smooth projective curve defined over an algebraic number field k, we investigate the finiteness of the set of generalized Jacobians of associated with modules defined over such that a fixed divisor representing a class of finite order in the Jacobian J of provides the torsion class in the generalized Jacobian . 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 , 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 for special elements of the field of functions of the hyperelliptic curve . [ABSTRACT FROM AUTHOR]