

International conference on Analytic Number Theory dedicated to 75th anniversary of G. I. Arkhipov and S. M. Voronin
December 14, 2020 13:15–13:45, Moscow, online






Functional continued fractions with large period lengths
G.V. Fedorov^{ab} ^{a} Scientific Research Institute for System Studies of RAS, Moscow
^{b} Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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Abstract:
The problem of periodicity of functional continued fractions of elements of a hyperelliptic field is closely related to the problem of finding and constructing the fundamental $S$units of a hyperelliptic field and the problem of torsion in the Jacobian of the corresponding hyperelliptic curve. For elliptic curves over the field of rational numbers, the torsion problem was solved by B. Mazur in 1978. For hyperelliptic curves of genus 2 and higher over the field of rational numbers, these three problems remain open.
Over the past 20 years, the theory of functional continued fractions has become a powerful arithmetic tool for investigating these problems. With the development of new methods in the theory of functional continued fractions, some classical problems acquired new aspects. In this regard, of particular interest are the results that differ significantly
from the traditional case of numeric continued fractions.
One of these results is raised by the problem of bounding for the period lengths
of functional continued fractions of elements of a hyperelliptic field.
The talk is dedicated to upper bounds for the period lengths
of key elements of hyperelliptic fields over number fields.
In the case when the hyperelliptic field is determined by a polynomial of odd degree,
the finite length of the period is trivially estimated
by twice the degree of the fundamental $S$unit.
A more interesting and complicated case is when a hyperelliptic field is given by a polynomial of even degree.
We will prove sharp upper bounds for the period lengths of functional continued fractions of elements of hyperelliptic fields over number fields $K$, depending only on the genus of the hyperelliptic field,
the degree of extension $[K: \mathbb{Q}]$
and the order of the torsion group of the Jacobian of the corresponding hyperelliptic curve.
^{*} Conference identificator: 947 3270 9056 Password: 555834

