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Heuristic study of class groups of number fields. (√Čtude heuristique des groupes de classes des corps de nombres.) (French) Zbl 0699.12016

We generalize the Cohen-Lenstra heuristics to class groups of arbitrary and relative extensions of number fields. For this, we need to develop combinatorial and algebraic tools concerning finite modules over semi- simple algebras. These theorems are of independent interest.
The heuristic results obtained are similar to those of Cohen and Lenstra; essentially, one must replace the Riemann zeta function by the zeta function of a suitable semisimple algebra. In addition, prime numbers dividing the degree of the Galois closure should a priori be excluded, but our predictions should also be valid for certain primes dividing this degree, although experimental evidence shows that one must be very careful with these primes.
The explicit numerical conjectures corresponding to some of the heuristic results of this paper have already been published by the authors [Ber. Math. Stat. Sekt. Forschungszent. Graz 272, 59–69 (1986; Zbl 0628.12008); Math. Comput. 48, 123–137 (1987; Zbl 0627.12006)].

MSC:

11R29 Class numbers, class groups, discriminants
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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