  Complex arithmetic random waves are stationary Gaussian complex-valued
solutions of the Helmholtz equation on the two-dimensional flat torus. We use
Wiener-It\^o chaotic expansions in order to derive a complete characterization
of the second order high-energy behaviour of the total number of phase
singularities of these functions. Our main result is that, while such random
quantities verify a universal law of large numbers, they also exhibit
non-universal and non-central second order fluctuations that are dictated by
the arithmetic nature of the underlying spectral measures. Such fluctuations
are qualitatively consistent with the cancellation phenomena predicted by Berry
(2002) in the case of complex random waves on compact planar domains. Our
results extend to the complex setting recent pathbreaking findings by Rudnick
and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci,
Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the
variance is based on a fine analysis of the Kac-Rice kernel around the origin,
as well as on a novel use of combinatorial moment formulae for controlling
long-range weak correlations.
