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Fundamental units for orders of unit rank 1 and generated by a unit

100%

Louboutin S.

Banach Center Publications

2016

tom 108

nr 1

173-189

Let ε be an algebraic unit for which the rank of the group of units of the order ℤ[ε] is equal to 1. Assume that ε is not a complex root of unity. It is natural to wonder whether ε is a fundamental unit of this order. It turns out that the answer is in general yes, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in ε) in the rare cases when the answer is no. This paper is a self-contained exposition of the solution to this problem, solution which was up to now scattered in many papers in the literature.

Simple proofs of the Siegel-Tatuzawa and Brauer-Siegel theorems

100%

Louboutin S.

Colloquium Mathematicum

2007

tom 108

nr 2

277-283

We give a simple proof of the Siegel-Tatuzawa theorem according to which the residues at s = 1 of the Dedekind zeta functions of quadratic number fields are effectively not too small, with at most one exceptional quadratic field. We then give a simple proof of the Brauer-Siegel theorem for normal number fields which gives the asymptotics for the logarithm of the product of the class number and the regulator of number fields.

Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

100%

Louboutin S.

Acta Arithmetica

2006

tom 121

nr 3

199-220

Note on a hypothesis implying the non-vanishing of Dirichlet L-series L(s,χ) for s > 0 and real characters χ

100%

Louboutin S.

Colloquium Mathematicum

2003

tom 96

nr 2

207-212

We prove that if χ is a real non-principal Dirichlet character for which L(1,χ) ≤ 1- log2, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ) > 0 for s > 0.

On the fundamental units of some cubic orders generated by units

63%

Lee J. , Louboutin S.

Acta Arithmetica

2014

tom 165

nr 3

283-299

Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field ℚ(ϵ) is Galois. Let ϵ, ϵ' and ϵ'' be the three real conjugates of ϵ. We tackle the problem of whether {ϵ,ϵ'} is a system of fundamental units of the cubic order ℤ[ϵ,ϵ',ϵ'']. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic polynomials are considered. We also improve upon and correct several previous results in the literature.