Arithmetic differential equations in several variables
We survey recent work on arithmetic analogues of ordinary and partial differential equations.
Arithmetic discriminants and horizontal intersections.
Arithmetic Distance Functions and Height Fuctions in Diophantine Geometry.
Arithmetic of formal groups and applications I: Universal norm subgroups.
Automorphisms and local rigidity of regular varieties
Beyond two criteria for supersingularity: coefficients of division polynomials
Let be a cubic, monic and separable polynomial over a field of characteristic and let be the elliptic curve given by . In this paper we prove that the coefficient at in the –th division polynomial of equals the coefficient at in . For elliptic curves over a finite field of characteristic , the first coefficient is zero if and only if is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the zero loci...
Canonical integral structures on the de Rham cohomology of curves
For a smooth and proper curve over the fraction field of a discrete valuation ring , we explain (under very mild hypotheses) how to equip the de Rham cohomology with a canonical integral structure: i.e., an -lattice which is functorial in finite (generically étale) -morphisms of and which is preserved by the cup-product auto-duality on . Our construction of this lattice uses a certain class of normal proper models of and relative dualizing sheaves. We show that our lattice naturally...
Classical and overconvergent modular forms
Classification of projective surfaces with small sectional genus : char >
Cohérence sur D+ et irrégularité des isocristaux surconvergents de rang 1.
Cohomology on affinoid spaces
Coleman integration versus Schneider integration on semistable curves.
Comparaison entre la cohomologie algebrique et la cohomologie p-adique rigide ä coefficients dans un module differentiel I. (Cas des courbes).
Comparaison entre la cohomologie algébrique et la cohomologie -adique rigide à coefficients dans un nodule différentiel
Component groups of abelian varieties and Grothendieck's duality conjecture
We investigate Grothendieck’s pairing on component groups of abelian varieties from the viewpoint of rigid uniformization theory. Under the assumption that the pairing is perfect, we show that the filtrations, as introduced by Lorenzini and in a more general way by Bosch and Xarles, are dual to each other. Furthermore, the methods yield some progress on the perfectness of the pairing itself, in particular, for abelian varieties with potentially multiplicative reduction.
Component groups of Néron models via rigid uniformization.
Congruenze per integrali ellittici
Construction analytique de courbes en géométrie non archimédienne
Construction analytique (rigide) de variétés abéliennes
Construction analytique rigide de variétés abéliennes