Christian Maire

Tamely Ramified Towers and Discriminant Bounds for Number Fields

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR2m. One knows that α0 ≥ 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 ≥ 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

Tamely ramified towers and discriminant bounds for number fields ---II

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R0 (2m) be the minimal root discrimniant for totally complex number fields of degree 2m, and put . Define R(m) to be the minimal root discriminant of totally real number fields of degree m and a1 = lim infm R1(m).

Sur les invariants d'Iwasawa des tours cyclotomiques

We carry out the computation of the Iwasawa invariants ρT , � T , λ T associated to abelian T-ramified S-decomposed l-extensions over the finite steps Kn of the cyclotomic Zl-extension K1/K of a number field of CM-type.

On the invariant factors of class groups in towers of number fields

For a finite abelian p-group A of rank d, we define its (logarithmic) mean exponent to be the base-p logarithm of the d-th root of its cardinality. We study the behavior of the mean exponent of p-class groups in towers of number fields. By combining techniques from group theory with the Tsfasman-Valdut generalization of the Brauer-Siegel Theorem, we construct infinite tamely ramified towers in which the mean exponent of class groups remains bounded. Several explicit examples with p=2 are given. We introduce an invariant M(G) attached to a finitely generated FAb pro-p group G which measures the asymptotic growth of the mean exponent of abelianizations of subgroups of index n with n going to infinity. When G=Gal(L/K), M(G) measures the asymptotic behavior of the mean exponent of class groups in L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

A propos de la tour localement cyclotomique d’un corps de nombres

We adapt some recent results on Hilbert ℓ-towers of number fields to locally cyclotomic ℓ-towers (i.e. logarithmic towers).Résumé. Nous adaptons des résultats récents sur le problème de la tour de Hilbert d’un corps de nombres au cas de la ℓ-tour localement cyclotomique.

On the Zℓ-rank of abelian extensions with restricted ramification [Journal of Number Theory 92 (2002) 376–404]

In the original Section 3.3, the main result is Theorem 29. In the proof of this Theorem, we use the fact that the cyclotomic extension k∞ is not contained in ∏p∈SlkSp (except for the trivial case N=Q). In fact, there is one other exception: the imaginary quadratic field. Moreover, there were some imprecisions in Proposition 28. In this note, we clarify all of this and we produce the complete proof of the key result.