Sur le 3-rang des corps quadratiques de discriminant premier ou presque premier, by Karim Belabas and Etienne Fouvry

The main result of this paper is the following statement: there exists a positive density of primes p (that can be chosen to be congruent to 1 or 3 mod 4, at will), such that the ideal class group of the real quadratic field $Q(\sqrt{p})$ has no elements of order 3. Analogous but weaker statements hold for imaginary quadratic fields: one obtains pseudo primes, not primes. If we insist on the 2-rank being 0, we only get r_3(Q(\sqrt{-p}) <= 1 The result is obtained by combining sieve methods with Davenport and Heilbronn's theory. This improves on the first author's earlier results by a careful estimate of exponential sums, treating various error terms on average, and considering "quasi-fundamental" discriminants instead of sieving for squarefree numbers.

Karim Belabas and Etienne Fouvry <karim@mpim-bonn.mpg.de, fouvry@matups.matups.fr>