Karim Belabas

A fast algorithm to compute cubic fields

We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to 10 11 and complex cubic fields down to -10 11 have been computed.

A relative van Hoeij algorithm over number fields

Abstract van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as the Berlekamp–Zassenhaus algorithm, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over number fields.

Error estimates for the Davenport-Heilbronn theorems

We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of 3-torsion elements in the class groups of quadratic fields. In addition, we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2-torsion elements in the class groups of cubic fields. These results prove analytic continuation of the related Dirichlet series to the left of the line R(s) = 1.

Generators and relations for K_2 O_F.

Tates algorithm for computing K_2 O_F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order---the latter, together with some structural results on the p-primary part of K_2 O_F due to Tate and Keune, gives a proof of its structure for many number fields of small discriminants, confirming earlier conjectural results. For the first time, tame kernels of non-Galois fields are obtained.

On quadratic fields with large 3-rank

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem, then analyze and improve the table-building algorithm. It computes the multiplicities of the O(X) general cubic discriminants (real or imaginary) up to X in time O(X) and space O(X), or more generally in time O(X +X/M) and space O(M +X) for a freely chosen positive M . A variant computes the 3-ranks of all quadratic fields of discriminant up to X with the same time complexity, but using only M + O(1) units of storage. As an application, we obtain the first 1618 real quadratic fields with r3(∆) > 4, and prove that Q( √ −5393946914743) is the smallest imaginary quadratic field with 3-rank equal to 5.

Counting Horoballs and Rational Geodesics

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. The asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M , are studied in this paper. The case of SL(2, ℤ), and of Bianchi groups, is developed.

Computing the residue of the Dedekind zeta function

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33 log D_K/(\sqrt{X}\log X), where D_K is the absolute value of the discriminant of K. Guided by Weils explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bachs constant to 2.33. This results in substantial speeding of one part of Buchmanns class group algorithm.

Corrigendum: on the mean 3-rank of quadratic fields

This corrigendum corrects a mistake in K. Belabas, On the mean 3-rank of quadratic fields , Compositio Math. 118 (1999), 1–9.

Computing Cubic Fields in Quasi-Linear Time

Cubic fields (over the rationals) are the simplest non-Galois number fields and thus should be the ideal testing ground for most general “density” conjectures, such as the Cohen-Martinet heuristics. We present an efficient algorithm to generate them, up to a given discriminant bound, which we hope will prove a useful tool in their computational exploration. It all originates from the seminal paper [4] by Davenport and Heilbronn and some reduction theory as was already known to Hermite. When no explicit reference has been given, we refer the curious reader not wishing to consider the proofs as (easy) exercises to [1]. The rationale is as follows: to a given cubic field, we associate first a class of binary cubic forms, which shares the same discriminant, and then a canonical representative in the class. The essential point is that we have an explicit description of the image of this mapping, the set of companion forms, which behaves nicely from the algorithmic point of view.