Computation of relative class numbers of imaginary abelian number fields.
Majoration au point 1 des fonctions L associées aux caractères de Dirichlet primitifs, ou au caractère d'une extension quadratique d'un corps quadratique imaginaire principal.
A finiteness theorem for imaginary abelian number fields.
Les extensions quadratiques dur corps des raionnels, ou du corps de Gauss, ou du corps des racines cubiques de l'unité des calibres 1.
Détermination des corps quartiques cycliques totalement imaginaires à groupe des classes d'idéaux d'exposant ...
Groupes des classes d'idéaux triviaux
Norme relative de l'unité fondamentale et 2-rang du groupe des classes d'idéaux de certains corps biquadratiques
Minoration au point 1 des fonctions L et détermination des corps sextiques abéliens totalement imaginaires principaux
The class number one problem for the dihedral and dicyclic CM-fields
We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.
The imaginary cyclic sextic fields with class numbers equal to their genus class numbers
It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic sextic fields with class numbers equal to their genus class numbers.
Corps quadratiques à corps de classes de Hilbert principaux et à multiplication complexe
The class number one problem for the non-abelian normal CM-fields of degree 16
Corps quadratiques à corps de classes de Hilbert principaux et à multiplication complexe
On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes
Lately, explicit upper bounds on (for primitive Dirichlet characters ) taking into account the behaviors of on a given finite set of primes have been obtained. This yields explicit upper bounds on residues of Dedekind zeta functions of abelian number fields taking into account the behavior of small primes, and it as been explained how such bounds yield improvements on lower bounds of relative class numbers of CM-fields whose maximal totally real subfields are abelian. We present here some other...
Explicit upper bounds for |L(1,χ)| for primitive even Dirichlet characters
The mean value of |L(k,χ)|² at positive rational integers k ≥ 1
Let k ≥ 1 denote any positive rational integer. We give formulae for the sums (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.
Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields
Simple proofs of the Siegel-Tatuzawa and Brauer-Siegel theorems
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.
Note on a hypothesis implying the non-vanishing of Dirichlet L-series L(s,χ) for s > 0 and real characters χ
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.
Fundamental units for orders of unit rank 1 and generated by a unit
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...
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