On Tsfasman--Vlăduţ Invariants of Infinite Global Fields

Quelques résultats effectifs concernant les invariants de Tsfasman-Vladuts

We consider properties of infinite algebraic extensions of global fields through their Tsfasman-Vladuts invariants (related in particular to the decomposition of primes). We use recent results of A. Schmidt and a weak effective version of the Grunwald-Wang theorem to construct infinite global fields having at the same time a given finite set of positive invariants, a prescribed set of invariants being zero and a controlled deficiency.

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…

On certain infinite families of imaginary quadratic fields whose Iwasawa λ-invariant is equal to 1

Let p be an odd prime number. In this paper, we show existence of certain infinite families of imaginary quadratic fields in which p splits and whose Iwasawa λ-invariant of the cyclotomic Zp-extension is equal to 1.

Theta invariants of euclidean lattices and infinite-dimensional hermitian vector bundles over arithmetic curves

In this monograph, we lay some foundations of a theory of infinite dimensional Euclidean lattices - and more generally, of infinite dimensional Hermitian vector bundles over some "arithmetic curve" ${\rm Spec}\,\mathcal{O}_K$ attached to the ring of integers $\mathcal{O}_K$ of some number field $K$ - with a view towards applications to transcendence theory and Diophantine geometry. In the first chapters of this monograph, we study the properties of the invariant $h^0_θ(\overline{E})$ attached to some Euclidean lattice $\overline{E}:= (E, \Vert.\Vert)$, defined by the expression $$h^0_θ(\overli…

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (Rényi). We show that they coincide exactly with two sets of quadratic irrationals that we call $E$-reduced, and respectively $B$-reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.

无穷多族量子模3-流形不变量

Infinite Families of Quantum Modular 3-Manifold Invariants

本文利用AJK理论扩展了3-流形的格上同调与BPS q-级数，首次为无穷多族负定plumbed 3-流形计算了AJK级数，并构造了可视为WRT不变量形变且具量子模性质的无穷多族不变量。