Zero cycles of degree one on principal homogeneous spaces

Implications of the Hasse Principle for Zero Cycles of Degree One on Principal Homogeneous Spaces

Let k be a perfect field of virtual cohomological dimension ≤ 2. Let G be a connected linear algebraic group over k such that Gsc satisfies a Hasse principle over k. Let X be a principal homogeneous space under G over k. We show that if X admits a zero cycle of degree one, then X has a k-rational point. Introduction The following question of Serre [10, p. 192] is open in general. Q: Let k be a ...

Homogeneous varieties - zero cycles of degree one versus rational points

Examples of projective homogeneous varieties over the field of Laurent series over p-adic fields which admit zero cycles of degree one and which do not have rational points are constructed. Let k be a field and X a quasi-projective variety over k. Let Z0(X) denote the group of zero cycles on X and deg : Z0(X) → Z the degree homomorphism which associates to a closed point x of X, the degree [k(x...

On the Geometry of Principal Homogeneous Spaces

Let k be an algebraically closed field, let π : X → B be an elliptic surface defined over k, and let XK be the generic fiber of π, which is an elliptic curve defined over the field K = k(B), the function field of B. If f : Y → B is a genus one fibration locally isomorphic to X (in the étale topology on B), then Y corresponds to a principal homogeneous space YK over XK which is everywhere locall...

On the Geometry of Principal Homogeneous Spaces

Let B be a curve defined over an algebraically closed field k and let π: X → B be an elliptic surface with base curve B. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for X, i.e., elements of the Tate-Shafarevich group of the generic fiber of π. If Y is such a principal homogeneous space of order n, we find strong restrictions on the Pn−1 bundle over B i...

Zero - Entropy Affine Maps on Homogeneous Spaces