Murmurations and Sato–Tate conjecture for high rank zetas of elliptic curves II: Beyond Riemann hypothesis

As a continuation of our earlier paper [Z. Shi and L. Weng, Murmurations and Sato–Tate conjectures for high rank zetas of elliptic curves, preprint, arXiv:2410.04952], 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 [Formula: see text]-invariant [Formula: see text] in rank [Formula: see text] even for the Sato–Tate law, rather, on a much refined structure, a similar version of which was already observed earlier by Zagier and the senior author of this paper in [L. Weng and D. Zagier, Higher rank zeta functions for elliptic curves, Proc. Natl. Acad. Sci USA 117(9) (2020) 4546–4558] when the rank [Formula: see text] Riemann hypothesis was established. Namely, instead of the rank [Formula: see text] Riemann hypothesis bounds [Formula: see text] on which our first paper is based, we use the asymptotic bounds [Formula: see text]. Accordingly, rank [Formula: see text] Sato–Tate law can be established and rank [Formula: see text] murmurations can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves.