Advanced Search

Let $X$ be a rationally connected algebraic variety, defined over a number field $k.$ We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k;$ (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree...

Let X be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\surd a,\surd b)/k}\left(x\right)=Q(t₁,...,tₘ)²$ over a number field k. We prove that: (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of X; (2) the analogue for rational points is also valid assuming Schinzel’s hypothesis.