On fundamental $S$-units and continued fractions constructed in hyperelliptic fields using two linear valuations

In this paper, for elements of hyperelliptic fields, the theory of functional continued fractions of generalized type associated with two linear valuations has been formulated for the first time. For an arbitrary element of a hyperelliptic field, the continued fraction of generalized type converges to this element for each of the two selected linear valuations of the hyperelliptic field. Denote by $S$ the set consisting of these two linear valuations. We find equivalent conditions describing the relationship between the quasi-periodicity of a continued fraction of generalized type, the existence of a fundamental $S$-unit, and the existence of a class of divisors of finite order in the divisor class group of a hyperelliptic field. The last condition is equivalent to the existence of a torsion point in the Jacobian of a hyperelliptic curve. These results complete the algorithmic solution of the periodicity problem in the Jacobians of hyperelliptic curves of genus two.