The finiteness theorems for generalized Jacobians with nontrivial torsion

The questions on the finiteness of the set of rational points on an algebraic curve over algebraic number fields are fundamental questions of arithmetic algebraic geometry. In my talk I shall discuss new questions about the finiteness of the set of generalized Jacobians of a curve $C$, defined over algebraic number field, that are associated to the modules $m$ such that a fixed class of finite order in the Jacobian of $C$ lifts to the torsion class in the generalized Jacobian $J_m$. On one hand, such set of generalized Jacobians with the property mentioned above is infinite. On the other hand, we obtained finiteness results under additional assumptions on the support of $m$ or on the structure of the subgroup $J_m$. These results were applied to the problem of quasi-periodicity of continued rational fractions constructed in the power series $k((1/x))$, for some special class of elements of the field of rational functions on the hyperelliptic curve $y^2 = f(x)$ over an algebraic number field. In particular, for any $n$ we established the finiteness of the set of polynomials $g(x)$ of degree bounded by $n$, for which the expansion of the elements $g(x) \sqrt{f(x)}$ in the continued fraction is periodic.