Generalizes Nagao conjecture to hyperelliptic curves, computes moments to obtain Jacobian rank 4g+2 over Q(T), and proves second-moment bias for some families.
The Sato-Tate conjecture and Nagao's conjecture
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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 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.
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math.NT 1years
2019 1verdicts
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Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture
Generalizes Nagao conjecture to hyperelliptic curves, computes moments to obtain Jacobian rank 4g+2 over Q(T), and proves second-moment bias for some families.