In this talk, we present some generalizations of Lagrange's periodicity theorem in the classical theory of continued fractions. The main idea is to use a 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'' a basis of F over the rationals, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group. By extending the above argument adelically, we also obtain a p-adelic continued fraction algorithm and its periodicity for imaginary quadratic irrationals.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/246