Geodesic
On the classification problem~for~polynomials~$f$ with a~periodic continued fraction expansion of~$\sqrt{f}$ in hyperelliptic fields
Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 972-1007

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The classical problem of the periodicity of continued fractions for elements of hyperelliptic fields has a long and deep history. This problem has up to now been far from completely solved. A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomials with coefficients in the field $\mathbb{Q}$ of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over $\mathbb{Q}$ whose square root has a periodic continued fraction expansion in the field $\mathbb{Q}((x))$ of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials $f$ with periodic continued fraction expansion of $\sqrt{f}$ in elliptic fields with the field of rational numbers as the field of constants.
Keywords: periodicity problem, continued fractions, elliptic curves, hyperelliptic fields, Jacobian variety, divisor class group, symbolic calculations, computer algebra. 
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     author = {V. P. Platonov and G. V. Fedorov},
     title = {On the classification problem~for~polynomials~$f$ with a~periodic continued fraction expansion of~$\sqrt{f}$ in hyperelliptic fields},
     journal = {Izvestiya. Mathematics },
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     language = {en},
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V. P. Platonov; G. V. Fedorov. On the classification problem~for~polynomials~$f$ with a~periodic continued fraction expansion of~$\sqrt{f}$ in hyperelliptic fields. Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 972-1007. http://geodesic.mathdoc.fr/item/IM2_2021_85_5_a6/