Fundamental S-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves

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In 2010, Platonov proposed a fundamentally new approach to the torsion problem in Jacobi varieties of hyperelliptic curves over the field of rational numbers. This new approach is based on the calculation of fundamental units in hyperelliptic fields. It was applied to prove the existence of torsion points of new orders. In the paper, the notion of the degree of an S-unit is introduced and a relationship between the degree of an S-unit and the order of the corresponding torsion point of the Jacobian of a hyperelliptic curve is established. A complete exposition of the new method and results obtained on the basis of this method is contained in [2].