On families of hyperelliptic curves over the field of rational numbers, whose Jacobian contains torsion points of given orders

G. V. Fedorov

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 322-340

Résumé

One of the pressing contemporary problems of algebra and number theory is the problem of the existence and searching for fundamental $S$-units in hyperelliptic fields. The problem of the existence and searching of $S$-units in hyperelliptic fields is equivalent the solvability of the norm equation — the functional Pell equation — with some additional conditions on the form of this equation and its solution. There is a deep connection between points of finite order in Jacobian variety (Jacobian) of hyperelliptic curve and nontrivial $S$-units of hyperelliptic field. This connection formed the basis of the algebraic approach proposed by V.P. Platonov to the well-known fundamental problem of boundedness of torsion in Jacobian varieties of hyperelliptic curves. For elliptic curves over a field of rational numbers, the torsion problem was solved by Mazur in the 1970s. For curves of genus 2 and higher over the field of rational numbers, the torsion problem turned out to be much more complicated, and it is far from its complete solution. The main results obtained in this direction include to the description of torsion subgroups of Jacobian varieties of specific hyperelliptic curves, and also to the description of some families of hyperelliptic curves of the genus $g \ge 2$. In this article, we have found a new method for studying solvability. functional norm equations giving a full description hyperelliptic curves over the field of rational numbers, whose Jacobian varieties possess torsion points of given orders. Our method is based on an analytical study of representatives finite order divisors in a divisor class group of degree zero and their Mumford representations. As an illustration of the operation of our method in this article, we directly found all parametric families of hyperelliptic curves of genus two over the field of rational numbers, whose Jacobian varieties have rational torsion points of orders not exceeding five. Moreover, our method allows us to determine which parametric family found this curve belongs, whose Jacobian has a torsion point of order not exceeding five.

Détail

Keywords: continued fractions, fundamental units, $S$-units, torsion in the Jacobians, hyperelliptic fields, divisors, divisor class group.