Picard modular forms and the cohomology of local systems on a Picard modular surface

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