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.Comment: 35 pages. This is the first draft of the article published in the Journal of Number Theory. I will soon update the article with the accepted manuscrip

We construct a new Eisenstein cocycle, called the Shintani-Barnes cocycle, which specializes in a uniform way to the values of the zeta functions of general number fields at positive integers. Our basic strategy is to generalize the construction of the Eisenstein cocycle presented in the work of Vlasenko and Zagier by using some recent techniques developed by Bannai, Hagihara, Yamada, and Yamamoto in their study of the polylogarithm for totally real fields. We also closely follow the work of Charollois, Dasgupta, and Greenberg. In fact, one of the key ingredients which enables us to deal with general number fields is the introduction of a new technique, called the "exponential perturbation", which is a slight modification of the Q-perturbation studied in their work.

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