The Sato-Tate conjecture and Nagao's conjecture

TL;DR

This paper demonstrates that the Sato-Tate conjecture, modeled via random matrices, implies Nagao's conjecture for specific families of elliptic and hyperelliptic curves, linking deep conjectures in number theory.

Contribution

It establishes a connection between the Sato-Tate and Nagao's conjectures, showing that the former implies the latter for certain curve families.

Findings

Sato-Tate conjecture implies Nagao's conjecture for specific curves

Link between random matrix models and elliptic surface ranks

Extension to hyperelliptic curves

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.