Abstract

A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , where c 2 R?0 is an absolute constant, and where the o(1)-function depends on the degree of F . 1 Introduction Computing the class group and the regulator of an algebraic number field F are two major tasks of algorithmic algebraic number theory. In the last decade, several regulator and class group algorithms have been suggested (e.g. [16],[17],[18],[3]). In [2] the problem of the computational complexity of those algorithms was adressed for the first time. This question was then studied in [2] in great detail. The theoretical results and the computational experience show that computing class groups and regulators is a very difficult problem. More precisely, it turns out that even under the a...

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