On periodicity of continued fractions in hyperelliptic function fields
Let \(F\in k[X]\) be a square-free polynomial of degree \(2p+2\) over a field k of odd characteristic, and let C be the nonsingular model of the curve \(y^ 2=F(x)\). Then C is a hyperelliptic curve of genus \(p.\) Consider the Pell equation of F, i.e. \(P^ 2-Q^ 2F=1\). The author gives necessary and sufficient conditions for the existence of a non-trivial solution \((P,Q)\in (k[X])^ 2\), \(Q\neq 0\), in terms of the quasi-periodicity of the continued fraction expansion of the form (\(\sqrt{F}+L)/M\) (where L, M are polynomials and M divides \(F-L^ 2)\).
scientific article; zbMATH DE number 4202465