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Rings of Low Rank |

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). [Back] |