相关论文: Points rationnels et groupes fondamentaux : applic…

Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of…

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…

If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its \'etale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective…

In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bella\"iche asked the same question for the special case of Abelian varieties. These…

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some…

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then…

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism…

Given a proper family of varieties over a smooth base, with smooth total space and general fibre, all over a finite field k with q elements, we show that a finiteness hypothesis on the Chow groups, CH_i, i=0,1,...,r, of the fibres in the…

For a proper (not necessarily smooth) variety over a finite field with q elements, Berthelot-Bloch-Esnault proved a trace formula which computes the number of rational points modulo q in terms of the Witt vector cohomology. We show the…

We prove that the cohomology groups of an etale Q_p-local system on a smooth proper rigid analytic space are finite-dimensional Q_p-vector spaces, provided that the base field is either a finite extension of Q_p or an algebraically closed…

We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic $p>5$. As a consequence, any klt Fano threefold over a finite field has a rational…

In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and…

We prove an algebraic version of a classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove…

If the $\ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For…

A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a…

Long ago, Fontaine formulated conjectures (now theorems) relating \'etale and de Rham cohomologies of algebraic varieties over $p$-adic fields. In an earlier work we have shown that pro-\'etale and de Rham cohomologies of analytic varieties…

We show that there is no simple congruence formula for the number of points of the mod p reduction of a regular model of a smooth proper variety defined over a local field with $\ell$-adic cohomology supported in codimension $\ge \kappa$.

We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field, and prove that every linear quotient of the fundamental group of such a variety is…

We study the homotopy fixed points under the Frobenius endomorphism on the stable $\mathbb A^1$-homotopy category of schemes in characteristic $p>0$ and prove a rigidity result for cellular objects in these categories after inverting $p$.…

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of…