  We study the nodal sets of eigenfunctions of the Laplacian on the standard
d-dimensional flat torus. The question we address is: Can a fixed hypersurface
lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In
dimension two, we show that this happens only for segments of closed geodesics.
In higher dimensions, certain cylindrical sets do lie on nodal sets
corresponding to arbitrarily large eigenvalues. Our main result is that this
cannot happen for hypersurfaces with nonzero Gauss-Kronecker curvature. In
dimension two, the result follows from a uniform lower bound for the L^2-norm
of the restriction of eigenfunctions to the curve, proved in an earlier paper.
In high dimensions we currently do not have this bound. Instead, we make use of
the real-analytic nature of the flat torus to study variations on this bound
for restrictions of eigenfunctions to suitable submanifolds in the complex
domain. In all of our results, we need an arithmetic ingredient concerning the
cluster structure of lattice points on the sphere. We also present an
independent proof for the two-dimensional case relying on the "abc-theorem" in
function fields.
