Colmez cones for fundamental units of totally real cubic fields (Q423610)

In 1976, \textit{T. Shintani} [J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 393--417 (1976; Zbl 0349.12007)] constructed, in connection with the zeta function of totally real number fields \(K\) with degree \(n\), a fundamental domain (consisting of certain cones) for the action of the totally positive units in \(K\) on \((0,\infty)^n\). Subsequent work by various authors provided explicit constructions of such cones in totally real cubic number fields, but all of them required an exhaustive search in suitable \(2\)-dimensional lattices. By using fundamental units of the type used by \textit{P. Colmez} [Invent. Math. 91, No. 2, 371--389 (1988; Zbl 0651.12010)], the authors succeed in finding a practical algorithm for constructing Shintani's cones for totally real cubic number fields.