Periodic elements $\sqrt{f}$ in elliptic fields with a field of constants of zero characteristic

Abstract

A study of the periodicity problem of functional continued fractions of elements of elliptic and hyperelliptic fields
was begun about 200 years ago in the classical papers of N.~Abel and P.~L.~Chebyshev.
In 2014 V.~P.~Platonov proposed a general conceptual method based on the deep connection between three classical problems:
the problem of the existence and construction of fundamental $S$-units in hyperelliptic fields,
the torsion problem in Jacobians of hyperelliptic curves,
and the periodicity problem of continued fractions of elements of hyperelliptic fields.
In 2015-2019, in the papers of V.~P.~Platonov et al. was made great progress in studying the problem of periodicity of elements in hyperelliptic fields,
especially in the effective classification of such periodic elements.
In the papers of V.~P.~Platonov et al, all elliptic fields $\mathbb{Q}(x)(\sqrt{f})$ were found
such that $\sqrt{f}$ decomposes into a periodic continued fraction in $\mathbb{Q}((x))$,
and also futher progress was obtained in generalizing the indicated result, as to other fields of constants,
and to hyperelliptic curves of genus $2$ and higher.
In this article, we provide a complete proof of the result announced by us in 2019
about the finiteness of the number of elliptic fields $k(x)(\sqrt{f})$ over an arbitrary number field $k$
with periodic decomposition of $\sqrt{f}$,
for which the corresponding elliptic curve contains a $k$-point of even order not exceeding $18$ or
a $k$-point of odd order not exceeding $11$.
For an arbitrary field $k$ being quadratic extension of $\mathbb{Q}$ all such elliptic fields are found,
and for the field $k = \mathbb{Q}$ we obtained new proof about of the finiteness of the number of periodic $\sqrt{f}$, not using the parameterization of elliptic curves and points of finite order on them.