  We strongly develop the relationship between complex hyperbolic geometry and
arithmetic counting or equidistribution applications, that arises from the
action of arithmetic groups on complex hyperbolic spaces, especially in
dimension $2$. We prove a Mertens' formula for the integer points over a
quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as
well as an equidistribution theorem of the set of rational points over $K$ in
Heisenberg groups. We give a counting formula for the cubic points over $K$ in
the complex projective plane whose Galois conjugates are orthogonal and
isotropic for a given Hermitian form over $K$, and a counting and
equidistribution result for arithmetic chains in the Heisenberg group when
their Cygan diameter tends to $0$.
