Geodesic
Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 1, pp. 1-34

Voir la notice de l'article provenant de la source Math-Net.Ru

In the past four years a theory has been developed for finding fundamental units in hyperelliptic fields, and on basis of this theory innovative and efficient algorithms for computing them have been constructed and implemented. A new local-global principle was discovered which gives a criterion for the existence of non-trivial units in hyperelliptic fields. The natural connection between the problem of computing fundamental units and the problem of torsion in Jacobian varieties of hyperelliptic curves over the rational number field has led to breakthrough results in the solution of this problem. The main results in the present survey were largely obtained using a symbiosis of deep theory, efficient algorithms, and supercomputing. Such a symbiosis will play an ever increasing role in the mathematics of the 21st century. Bibliography: 27 titles.
Keywords: fundamental units, hyperelliptic fields, local-global principle, Jacobian varieties, hyperelliptic curves, fast algorithms, continued fractions.
Mots-clés : torsion problem in Jacobians 
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V. P. Platonov. Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 1, pp. 1-34. http://geodesic.mathdoc.fr/item/RM_2014_69_1_a0/