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

Abstract

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 f >= 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.