The theory of functional continued fractions and the torsion problem in the Jacobians of hyperelliptic curves

One of the fundamental problems in number theory and algebraic geometry is the torsion problem in Jacobians (Jacobi varieties) of hyperelliptic curves over field of rational numbers. This problem has a long history dating back to the 19th century. For elliptic curves, the Jacobian is isomorphic to the curve itself. In the elliptic case with the field of constants $\mathbb{Q}$ the torsion problem was completely solved by Mazur in 1978.
In 2010, Academician V. P. Platonov proposed a new approach, based on a deep and natural connection of three problems:
– the problem of describing points of finite order in the Jacobians of hyperelliptic curves,
– the problem of describing fundamental S-units in hyperelliptic fields and
– the problem of periodicity of functional continued fractions of elements of hyperelliptic fields.
The talk will provide basic background information and some recent results in this area.