On periodicity of geodesic continued fractions

TL;DR

This paper generalizes classical continued fraction periodicity results using geometric interpretations related to geodesics, extending to number fields and p-adic contexts, revealing deep connections with unit groups.

Contribution

It introduces a geodesic multi-dimensional continued fraction algorithm for number field bases and proves its periodicity, linking periods to relative unit groups, and extends to p-adic cases.

Findings

Constructed a geodesic multi-dimensional continued fraction algorithm.

Proved the periodicity of the algorithm and its relation to unit groups.

Developed a p-adic continued fraction algorithm with proven periodicity.

Abstract

In this paper, we present some generalizations of Lagrange's theorem in the classical theory of continued fractions motivated by the geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension $F/F'$ of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to "expand" any basis of $F$ over $\mathbb{Q}$, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group. By developing the above argument adelically, we also obtain a $p$-adelic continued fraction algorithm and its periodicity for imaginary quadratic irrationals.