On the classification problem for polynomials with a periodic continued fraction expansion of in hyperelliptic fields

Abstract

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 of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over whose square root has a periodic continued fraction expansion in the field of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials with periodic continued fraction expansion of in elliptic fields with the field of rational numbers as the field of constants.