We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the analogous problem over Q. As an application we show that any smooth hypersurface over K whose dimension is large enough in terms of the degree is K-unirational, provided that either the degree is odd or K is totally imaginary.
The analytic Hasse Principle for certain singular intersections of quadrics in $\mathbb{P}^9$
Forms representing forms and linear spaces on hypersurfaces
Period and index of genus one curves over number fields
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