On the Parametrization of Hyperelliptic Fields with $S$-Units of Degrees~7 and 9

G. V. Fedorov ; V. S. Zhgoon ; M. M. Petrunin ; Yu. N. Shteinikov

Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 444-452.

Résumé

We show that if $k$ is an algebraically closed field with $\operatorname{char}k=0$, then the set of polynomials $f$ of degree $5$ such that the field $k(x)(\sqrt{f}\,)$ has a nontrivial $S$-unit of degree $7$ or $9$ and the continued fraction expansion of $\sqrt{f}/x$ is periodic is a one-parameter set corresponding to a rational curve with finitely many deleted points.

Détail

Keywords: hyperelliptic field, rational curve, Gröbner basis.
Mots-clés : torsion point

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