Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.

Si un système d’équations polynomiales à coefficients entiers admet une solution dans ${𝐐}^n$, il en admet sur tout complété $p$-adique ou réel de $𝐐$. La réciproque a été démontrée par Hasse pour les quadriques, mais elle est fausse en général. Une grande partie des contre-exemples connus peuvent être expliqués à l’aide de l’obstruction de Brauer-Manin, basée sur la théorie du corps de classe. Il est donc naturel de se demander si, pour certaines classes de variétés, cette obstruction est la seule. Le but...

Let $C_m:y^2=x^3-m^{2}x+p^2{q}^2$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p$, $q$ are distinct odd primes. We study the torsion part and the rank of $C_m\left(\mathbb{Q}\right)$. More specifically, we prove that the torsion subgroup of $C_m\left(\mathbb{Q}\right)$ is trivial and the $\mathbb{Q}$-rank of this family is at least 2, whenever $m\neg \equiv 0(mod3)$, $m\neg \equiv 0(mod4)$ and $m\equiv 2(mod64)$ with neither $p$ nor $q$ dividing $m$.

We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for $\ell$-adic Lie extensions of a separated scheme $X$ of finite type over a finite field of characteristic prime to $\ell$.

In this paper we give a characterization of the height of K3 surfaces in characteristic $p>0$. This enables us to calculate the cycle classes in families of K3 surfaces of the loci where the height is at least $h$. The formulas for such loci can be seen as generalizations of the famous formula of Deuring for the number of supersingular elliptic curves in characteristic $p$. In order to describe the tangent spaces to these loci we study the first cohomology of higher closed forms.

We study applications of divisibility properties of recurrence sequences to Tate’s theory of abelian varieties over finite fields.

Let $m\in {\mathbb{Z}}_{>0}$ and a,q ∈ ℚ. Denote by ${}_m(a,q)$ the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve $E_d:x²+y²=1+dx²y²$. We study the set ${}_m(a,q)$ and we parametrize it by the rational points of an algebraic curve.