The rank of étale cohomology of varieties over -adic or number fields
The rationality of the Poincaré series associated to the p-adic points on a variety.
The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality.
The reduction of a double covering of a Mumford curve.
The Riemann-Roch theorem and geometry, 1854-1914.
The Schottky-Jung theorem for Mumford curves
The Schottky-Jung proportionality theorem, from which the Schottky relation for theta functions follows, is proved for Mumford curves, i.e. curves defined over a non-archimedean valued field which are parameterized by a Schottky group.
The Shafarevich-Tate Conjecture for Pencils of Elliptic Curves on K3 Surfaces.
The sign of the functional equation of the L-function of an orthogonal motive.
The size function h° for a pure cubic field
The solubility of diagonal cubic surfaces
The Strong Lefschetz Principle in Algebraic Geometry.
The structure of proper rigid groups.
The tame fundamental group of an abelian variety and integral points
The Tate conjecture for ordinary K3 surfaces over finite fields.
The topology of rational points.
The two faces of the twisted Kummer surface
The Uniform Position Principle for Curves in Characteristic p.
The universal tropicalization and the Berkovich analytification
Given an integral scheme over a non-archimedean valued field , we construct a universal closed embedding of into a -scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of by previous work of the authors, and we show that the set-theoretic tropicalization of with respect to this universal embedding is the Berkovich analytification . Moreover, using the scheme-theoretic...
The universal vectorial Bi-extension and p-adic heights.
The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces
We study the functional codes defined on a projective algebraic variety , in the case where is a non-degenerate Hermitian surface. We first give some bounds for , which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code .