Trace formulas for a class of compact complex surfaces

We give the trace formulas of weight $k$ for cocompact, torsion-free discrete subgroups of $SU(2, 1)$ and prove the analogue of the Riemann hypothesis on compact complex surfaces $M$ with $c_1^2(M)=3 c_2(M)$, where $c_i(M)$ is the $i$-th Chern class of $M$, $c_2(M)$ is a multiple of three and $c_2(M)>0$.