Calculation of the fundamental $S$-units in hyperelliptic fields of genus $2$ and the torsion problem in the jacobians of hyperelliptic curves

M. M. Petrunin

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 250-283

Résumé

A new approach to the torsion problem in the Jacobians of hyperelliptic curves over the field of rational numbers was offered by Platonov. This new approach is based on the calculation of fundamental units in hyperelliptic fields. The existence of torsion points of new orders was proved with the help of this approach. The full details of the new method and related results are contained in [2]. Platonov conjectured that if we consider the $S$ consisting of finite and infinite valuation and change accordingly definition of the degree of $S$-unit, the orders of torsion $\mathbb Q$-points tend to be determined by the degree of fundamental $S$-units. The main result of this article is the proof of existence of the fundamental $S$-units of large degrees. The proof is based on the methods of continued fractions and matrix linearization based on Platonov's approach. Efficient algorithms for computing $S$-units using method of continued fractions have been developed. Improved algorithms have allowed to construct the above-mentioned fundamental $S$-units of large degrees. As a corollary, alternative proof of the existence of torsion $\mathbb Q$-points of some large orders in corresponding Jacobians of hyperelliptic curves was obtained. Bibliography: 19 titles.

Détail

Keywords: fundamental unit, $S$-unit, hyperelliptic fields, Jacobian, hyperelliptic curves, torsion problem in Jacobians, fast algorithms, continued fractions, matrix linearization, torsion $\mathbb Q$-points.