Abstract
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 to smooth irreducible projective surfaces. We show that the Sato-Tate conjecture based on the random matrix model implies Nagao's conjecture for surfaces which form families of quadratic twists of fixed elliptic curves or fixed hyperelliptic curves of genus g≥1.
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