Lorentz Center - Stieltjes Onderwijsweek Rings of Low Rank from 6 Jun 2006 through 9 Jun 2006
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    Stieltjes Onderwijsweek
    Rings of Low Rank

    from 6 Jun 2006 through 9 Jun 2006

Karim Belabas

Karim Belabas


Cubic rings: theory, practice and applications.


Abstract:  The goal of these lectures is to present a self-contained treatment of the theory of cubic rings, the first non-trivial number rings.  It is easy to parametrize them (Delone-Fadeev), and the parametrization repects ramification, in particular the discriminant, which leads to a wealth of interesting results. Obviously, one can build tables of cubic rings or fields ordered by discriminant (in essentially linear time in the output size). We also get asymptotic estimates for the number of cubic rings, orders or fields with bounded discriminants (theorems of Davenport-Heilbronn and B.-Fouvry).  Since the Galois closure of a cubic field of discriminant D is a cyclic cubic extension of Q(\sqrt{D}), we also get information on the 3-torsion of the ray class groups of the latter fields. Imposing various local conditions lead to probabilistic statements confirming predictions of Cohen-Lenstra type.


Even with such complete control, intriguing open problems remain. For instance, a prime discriminant is congruent to 1 mod 4 by Stickelberger's congruence. We do not know whether infinitely many cubic discriminants lie among these primes. On the other hand, a positive density of these primes are not cubic discriminants, but no explicit family is known. The main reason for our difficulties is a question that has attracted a lot of attention in the recent years: although it is uniformly bounded on average (Davenport-Heilbronn), it is hard to bound non-trivially the 3-torsion of the class group of Q(\sqrt{D}), in terms of the discriminant D (Peirce, Helfgott-Venkatesh, Ellenberg-Venkatesh).