Let $G$ be a complex, semisimple, simply connected algebraic group with Lie algebra ${\frak g}$. We extend scalars to the power series field in one variable ${\rm C}((\pi))$, and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of ${\frak g} \otimes {\rm C}((\pi))$, i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig's topological construction of Springer's representations to the affine context.

We study a family of K3 surfaces which have a big automorphism group. We begin with generalisations of Silverman's results: construction of canonical heights, density of rational points in one orbit,… We continue the study in estimating the density of rational points on the orbiting rational curves; this estimate is compatible with Batyrev–Manin conjecture. Moreover we settle, under more geometric hypothesis, the number of rational points of such surfaces of bounded height.

We proove an analogue of the Beilinson-Bernstein Theorem, in the case of the projective space, for the ring of arithmetical differential operators introduced by P. Berthelot.