Calculation of the class numbers and fundamental units of abelian extensions over imaginary quadratic fields from approximate values of elliptic units (Q1074654)

Let F be \({\mathbb{Q}}\) or an imaginary quadratic field, and let L be an abelian extension of F (with the condition L real if \(F={\mathbb{Q}})\). The author gives a general procedure to compute the class number h of L and finds simultaneously the fundamental units of L. When \(F={\mathbb{Q}}\), the principle of calculation has been introduced by \textit{G. Gras} and the reviewer [Bull. Sci. Math., II. Sér. 101, 97-129 (1977; Zbl 0359.12007)]. When F is an imaginary quadratic field, the author uses the elliptic units instead of the cyclotomic ones, and gives an explicit procedure for a computer. As in the absolute abelian case, the method consists in computing an explicit upper bound for the index of elliptic units in units of L, and in deducing this index from an approximate value of the elliptic units. The author gives some numerical examples, and has applied this algorithm particularly in the cubic, quartic and sextic cases [Proc. Japan Acad., Ser. A 57, 56-59, 117-120, 363-366 (1981; Zbl 0478.12003, Zbl 0478.12004, Zbl 0515.12003)].