  We formulate a detailed conjectural Eichler-Shimura type formula for the
cohomology of local systems on a Picard modular surface associated to the group
of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is
based on counting points over finite fields on curves of genus three which are
cyclic triple covers of the projective line. Assuming the conjecture we are
able to calculate traces of Hecke operators on spaces of Picard modular forms.
We provide ample evidence for the conjectural formula.
  Along the way we prove new results on characteristic polynomials of Frobenius
acting on the first cohomology group of cyclic triple covers of any genus,
dimension formulas for spaces of Picard modular forms and formulas for the
numerical Euler characteristics of the local systems.
