On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields

Abstract

For a prime ideal P of the ring of integers of a number field K, we give a general definition of P-adic continued fraction, which also includes classical definitions of continued fractions in the field of p-adic numbers. We give some necessary and sufficient conditions on K ensuring that every α ∈ K admits a finite P-adic continued fraction expansion for all but finitely many P, addressing a similar problem posed by Rosen in the archimedean setting.

Quaternionic $p$-adic continued fractions

“…Following the terminology introduced in [MVV20] in the real case and in [CMT21], we shall say that a quaternionic type τ satisfies the Quaternionic Continued Fraction Finiteness (QCFF) property if every α ∈ B has a finite expansion of type τ . We say that a pair (B, R) has the p-adic QCFF property if there exists a quaternionic type (B, R, p, s) enjoying the QCFF property.…”

“…While the idea of defining continued fractions over quaternions was already suggested by Hamilton [Ham], a standard terminology for the p-adic case is still missing. Therefore, in analogy with the case of number fields [CMT21], in Section 3 we define the notion of quaternionic type associated to an order R in B. Namely, this is a quadruple τ = (B, R, p, s) where B is a quaternion algebra, R is an order in B, p a prime ≥ 3 and s a "p-adic floor function" taking values in R[ 1 p ]. Each quaternionic type gives rise to an algorithm that computes the continued fraction expansion of every element in the p-adic completion of B.…”

“…In Section 5, we use Northcott property to prove Theorem 5.2, which provides a criterion to establish the finiteness of the continued fraction algorithm in some cases. Adopting a similar terminology as in [CMT21], we say that a type τ satisfies the Quaternionic Continued Fraction Finiteness (QCFF) if every α ∈ B has a finite continued fraction expansion of type τ . Section 6 deals with indefinite quaternion algebras of discriminant pq: in this case, we give an explicit expression for a maximal order, and we construct some concrete examples of quaternionic types for which the QCFF does not hold.…”