Linear Spaces on Hypersurfaces over Number Fields

Abstract

We establish an analytic Hasse principle for linear spaces of affine dimension $m$ on a complete intersection over an algebraic field extension $\mathbb{K}$ of $\mathbb{Q}$. The number of variables required to do this is no larger than what is known for the analogous problem over $\mathbb{Q}$. As an application, we show that any smooth hypersurface over $\mathbb{K}$ whose dimension is large enough in terms of the degree is $\mathbb{K}$-unirational, provided that either the degree is odd or $\mathbb{K}$ is totally imaginary.