We prove lower and upper bounds for the class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta functions.

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜=\mathbb{Q}(\sqrt{2pq},\mathrm{i})$, where $\mathrm{i}=\sqrt{-1}$ and $p\equiv -q\equiv 1(mod4)$ are different primes. For each of the three quadratic extensions $𝕂/𝕜$ inside the absolute genus field ${𝕜}^{(*)}$ of $𝕜$, we determine a fundamental system of units and then compute the capitulation kernel of $𝕂/𝕜$. The generators of the groups ${\mathrm{Am}}_s(𝕜/F)$ and $\mathrm{Am}(𝕜/F)$ are also determined from which we deduce that ${𝕜}^{(*)}$ is smaller than the relative genus field ${(𝕜/\mathbb{Q}\left(\mathrm{i}\right))}^{*}$. Then we prove that each...

Let $K$ be an imaginary cyclic quartic number field whose 2-class group is of type $(2,2,2)$, i.e., isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. The aim of this paper is to determine the structure of the Iwasawa module of the genus field $K^{(*)}$ of $K$.

From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_K$, we study any such issue for arbitrary number fields $K$. We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$-adic obstructions coming from the global units of $K$. By restriction to the $p$-Sylow subgroups of $A_K$ and assuming the Leopoldt conjecture we show that the...

Let $K={𝔽}_q\left(T\right)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f\left(T\right)\in {𝔽}_q\left[T\right]$ with positive degree, denote by ${\Lambda }_f$ the torsion points of the Carlitz module for the polynomial ring ${𝔽}_q\left[T\right]$. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K\left({\Lambda }_P\right)$ of degree $k$ over ${𝔽}_q\left(T\right)$, where $P\in {𝔽}_q\left[T\right]$ is an irreducible polynomial of positive degree and $k>1$ is a positive divisor of $q-1$. A formula for the analytic class number for the...

Lately, explicit upper bounds on $\left|L\right(1,\chi \left)\right|$ (for primitive Dirichlet characters $\chi$) taking into account the behaviors of $\chi$ 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...

We compute the numbers of locally principal ideals with given norm in a class of definite quaternion orders and the traces of the Brandt-Eichler matrices corresponding to these orders. As an application, we compute the numbers of representations of algebraic integers by the norm forms of definite quaternion orders with class number one as well as we obtain class number relations for some CM-fields.

Let $p\equiv 1(mod8)$ and $q\equiv 3(mod8)$ be two prime integers and let $\ell \notin \{-1,p,q\}$ be a positive odd square-free integer. Assuming that the fundamental unit of $\mathbb{Q}\left(\sqrt{2p}\right)$ has a negative norm, we investigate the unit group of the fields $\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-\ell })$.

La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant $D$, sous l’action de ${\mathrm{SL}}_2$ par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant $D$. Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers,...