Periodicity of P-adic continued fraction expansions

Abstract

In 1940, K. Mahler presented a geometric algorithm which, for any P-adic integer ζ, yields a sequence of pairs of integers ( p n , q n ) which give P-adic approximations that are best with respect to Φ( X, Y), a real, reduced, positivedefinite quadratic form of determinant −1. The algorithm also constructs a sequence of 2 × 2 integer matrices of determinant P, denoted Ω(ζ), which defines the pairs ( p n , q n ) from a product of the first n matrices of Ω(ζ). In this paper, an equivalence relation is considered which relates P-adic integers ζ and ξ if their sequences Ω(ζ) and Ω(ξ) eventually agree. This is shown to happen if and only if ζ = Tξ, where T is an integer transformation of determinant P g which satisfies specified conditions. Mahler showed that if Ω(ζ) is periodic then ζ is rational or a quadratic irrational, yet for such a quadratic irrational, Φ can be chosen so that Ω(ζ) is no longer periodic. In this paper, the quadratic irrationals ζ, for which Ω(ζ) is periodic for some choice of Φ, are characterized. A relation similar to equivalence is used in the proof. In particular, it is concluded that there are quadratic irrationals ζ for which Ω(ζ) is never periodic for any Φ.