On S-Units for Linear Valuations and the Periodicity
of Continued Fractions of Generalized Type in Hyperelliptic Fields
Academician of the RAS V. P. Platonova,b,* and G. V. Fedorova,c,**
a Scientific Research Institute for System Analysis,
Russian Academy of Sciences, Moscow, 117218 Russia
b Steklov Mathematical Institute, Russian Academy
of Sciences, Moscow, 119991 Russia
c Faculty of Mechanics and Mathematics,
Lomonosov Moscow State University, Moscow, 119991 Russia
Correspondence to: *e-mail: platonov@niisi.ras.ru
Correspondence to: **e-mail: fedorov@mech.math.msu.su
Received 23 January, 2019
Abstract—An equivalence theorem is proved for the following conditions: the periodicity of continued fractions of generalized type for key elements of hyperelliptic field $null$, the existence of nontrivial $null$-units in $null$ for sets $null$ consisting two valuations of degree one, and the existence of a torsion of certain type in the Jacobian variety associated with hyperelliptic field $null$. In practice, this theorem allows using continued fractions of generalized type to effectively search for fundamental $null$-units of hyperelliptic fields. We give an example of the hyperelliptic field of genus 3, which shows all three equivalent conditions in the indicated theorem.
DOI: 10.1134/S1064562419030116