  Nagao's conjecture relates the rank of an elliptic surface to a limit formula
arising from a weighted average of fibral Frobenius traces, and it is further
generalized for smooth irreducible projective surfaces by M. Hindry and A.
Pacheco. We show that the Sato-Tate conjecture based on the random matrix model
implies Nagao's conjecture for certain twist families of elliptic curves and
hyperelliptic curves.
