Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis

Abstract

As a continuation of our earlier paper, we offer a new approach to murmurations and Sato-Tate laws for higher rank zetas of elliptic curves. Our approach here does not depend on the Riemann hypothesis for the so-called a-invariant in rank n>2 even for the Sato-Tate law, rather, on a much refined structure, a similar version of which was already observed by Zagier and the senior author when the rank n Riemann hypothesis was established. Namely, instead of the rank n Riemann hypothesis bounds, we use much stronger asymptotic bounds. Accordingly, rank n Sato-Tate law can be established and rank n murmuration can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves.