Xavier-François Roblot

On the computation of all extensions of a p -ADIC field of a given degree

Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K/k of a given degree and discriminant.

The Brumer-Stark conjecture in some families of extensions of specified degree

As a starting point, an important link is established between Brumers conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if K/F is an abelian extension of relative degree 2p, p an odd prime, we prove the l-part of the Brumer-Stark conjecture for all odd primes l ≠ p with F belonging to a wide class of base fields. In the same setting, we study the 2-part and p-part of Brumer-Stark with no special restriction on F and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.

Computing the Hilbert class field of real quadratic fields

Using the units appearing in Starks conjectures on the values of L-functions at s=0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field.

Stark's conjectures and Hilbert's twelfth problem

We give a constructive proof of a theorem of Tate, which states that (under Starks Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real abelian extensions of K. We give two examples.

Polynomial factorization algorithms over number fields

Abstract The aim of this paper is to describe two new factorization algorithms for polynomials. The first factorizes polynomials modulo the prime ideal of a number field. The second factorizes polynomials over a number field.

Numerical Evidence Toward a 2-adic Equivariant “Main Conjecture”

We test a conjectural nonabelian refinement of the classical 2-adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting.

Computing p-adic L-functions of totally real number fields

We prove explicit formulas for the p-adic L-functions of totally real number fields and show how these formulas can be used to compute values and representations of p-adic L-functions.

Numerical Verification of the Brumer-Stark Conjecture

The construction of group ring elements that annihilate the ideal class groups of totally complex abelian extensions of ℚ is classical and goes back to work of Kummer and Stickelberger. A generalization to totally complex abelian extensions of totally real number fields was formulated by Brumer. Brumer’s formulation fits into a more general framework known as the Brumer-Stark conjecture. We will verify this conjecture for a large number of examples belonging to an extended class of situations where the general status of the conjecture is still unknown.

Corrigendum to “The Brumer-Stark conjecture in some families of extensions of specified degree”

Barry Smith has found an error in the statement and proof of Lemma 2.5 in our paper [GRT] (Math. Comp. 73 (2004), 297-315). This Lemma concerns a cyclic Galois extension K/E of CM fields of odd prime degree p. Towards the end of the proof, it is claimed that every root of unity in E is a norm from K. Our reasoning for this has a gap (the local part of the argument does not work at places where the quadratic extension E/E is split, where F = E is the maximal totally real subfield of E), and the statement can indeed fail, as confirmed by a concrete example calculated by Barry Smith. We will first state a corrected version of the Lemma, and then we will explain how the proof of Proposition 2.2 has to be adapted. Fortunately the statement of this Proposition (please refer to [GRT]) need not be changed at all, and therefore the rest of the paper is unaffected. (The only other place where Lemma 2.5 is used is in the proof of Proposition 2.1, but there ζp is not in K, so the relevant case of the Lemma is case (i) below, where the formula remains the same.)

Numerical verification of the Stark-Chinburg conjecture for some icosahedral representations

In this paper, we give 14 examples of icosahedral representations for which we have numerically verified the Stark-Chinburg conjecture.