Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves

TL;DR

This paper extends the Sato-Tate conjecture and murmurations phenomena to non-abelian high-rank zeta functions of elliptic curve reductions, supported by computational evidence.

Contribution

It demonstrates that both murmurations and the Sato-Tate conjecture apply to non-abelian high-rank zeta functions of elliptic curves, broadening their scope.

Findings

Murmurations observed in non-abelian high-rank zeta functions.

Sato-Tate conjecture holds for these non-abelian zeta functions.

Computational evidence supports the conjectures in this new setting.

Abstract

For elliptic curves over rationals, there are a well-known conjecture of Sato-Tate and a new computational guided murmuration phenomenon, for which the abelian Hasse-Weil zeta functions are used. In this paper, we show that both the murmurations and the Sato-Tate conjecture stand equally well for non-abelian high rank zeta functions of the p-reductions of elliptic curves over rationals.