The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalises the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.

Determining the algebraic K-theory of rings of integers in number fields has been the goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q0(S ) → BGL(Z) (see 1.1 for the notation) induces an interesting map on homotopy groups from the stable homotopy groups of spheres to the algebraic K-theory of the ring Z of rational integers. Quillen observed that if ` is an odd prime...

Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain ...

Let G be a connected reductive algebraic group over an algebraic closed field. We define a (surjective) map from the set of conjugacy classes in the Weyl group to the set of unipotent classes in G.

In addition, define certain lattices and subsets: L∨ = X∗(T ) = group of algebraic homomorphisms Gm → T L = Hom(X∗(T ),Z), the dual of L ∨ X(T ) = group of algebraic homomorphisms T → Gm, the weights of T . There is a canonical map from L toX(T )—to λ corresponds the multiplicative character e defined by the formula e(μ(x)) = x 〉 , which makes sense because all algebraic homomorphisms Gm → Gm a...

Let f : CP 99K CP be a rational map with algebraic and topological degrees both equal to d ≥ 2. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms T : CP → CP, the perturbed map T ◦f admits exactly two ergodic measures of maximal entropy log d, one of saddle and one of repelling type. Neither measure is supported in...

One of the best known discrete integrable systems is the billiard inside an n-dimensional ellipsoid. Veselov showed that generic complex invariant manifolds of the algebraic billiard map are open subsets of (coverings of) hyperelliptic Jacobians and that the restriction of the map to such manifolds is described by the shift by a constant vector (an algebraic addition law). Later, Dragovic and R...

We show how to express a conformal map Φ of a general two connected domain in the plane such that neither boundary component is a point to a representative domain of the form Ar = {z : |z + 1/z| < 2r}, where r > 1 is a constant. The domain Ar has the virtue of having an explicit algebraic Bergman kernel function, and we shall explain why it is the best analogue of the unit disc in the two conne...