An effective criterion for periodicity of $\ell $-adic continued fractions

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The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. For example, in the l-adic case, rational numbers may have a periodic non-terminating expansion; moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction expansion is periodic seems to be still open. In the present paper we investigate the l-adic continued fraction expansions of rationals and quadratic irrationals using the definition introduced by Ruban. We give general explicit criteria to establish the possible periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol).