  We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on
the three-dimensional standard flat torus, in particular the asymptotic
distribution of the nodal intersection length against a fixed regular reference
surface. Expectation and variance have been addressed by Maffucci (2019) who
found that the expected length is proportional to the square root of the
eigenvalue times the area of the surface, while the asymptotic variance only
depends on the geometry of the surface, the projected lattice points being
equidistributed on the two-dimensional unit sphere in the high-energy limit. He
also noticed that there are ``special'' surfaces, so-called static, for which
the variance is of smaller order; however he did not prescribe the precise
asymptotic law in this case. In this paper, we study second order fluctuations
of the nodal intersection length. Our first main result is a Central Limit
Theorem for ``generic'' surfaces, while for static ones, a sphere or a
hemisphere e.g., our main results are a non-Central Limit Theorem and a precise
asymptotic law for the variance of the nodal intersection length, conditioned
on the existence of so-called well-separated sequences of Laplacian
eigenvalues. It turns out that, in this regime, the nodal area investigated by
Cammarota (2019) is asymptotically fully correlated with the length of the
nodal intersections against any sphere. The main ingredients for our proofs are
the Kac-Rice formula for moments, the chaotic decomposition for square
integrable functionals of Gaussian fields, and some arithmetic estimates that
may be of independent interest.
