On the unboundedness of period lengths of functional continued fractions in a hyperelliptic field

The report is devoted to joint results with V. P. Platonov concerning the problem of the unboundedness of period lengths of continued fractions of elements from a hyperelliptic field. The famous Abel theorem establishes a criterion for the existence of elements in a hyperelliptic field that have a periodic expansion into a continued fraction. Subsequently, a significant number of studies were aimed at studying the problem of periodicity of functional continued fractions, including obtaining upper bounds on possible period lengths. Until now, the problem of the finiteness of the set of possible period lengths of continued fractions for a given hyperelliptic field has remained open. In our report, we will present results that give a negative solution to this problem.