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

Zhan Shi, Lin Weng

TL;DR

This work extends murmurations and Sato-Tate phenomena from rank-one to rank-$n$ zetas of elliptic curves, addressing higher-rank distributions without relying on the rank $n$ Riemann Hypothesis. It introduces a refined, RH-free framework built on sharp asymptotics for rank-$n$ invariants, notably the $a_{E/\mathbb{F}_q;n}$ and associated $\Delta$-type observables, and demonstrates that rank $n$ Sato-Tate laws can be observed via new $\Theta$-distributions that align with the classical sine-squared measure. A key technical development is the recasting of rank-$n$ observations through $\Delta'$-distributions and a direct connection to the conventional Sato-Tate law, as well as the introduction of new murmuration functionals $f_{r,n}^{\text{new}}$ that reveal rank $n$ stability in families of elliptic curves. The results imply that higher-rank arithmetic structures underlying the murmurations are canonically linked to the familiar abelian Sato-Tate statistics, with practical implications for understanding non-abelian zeta phenomena in arithmetic geometry.

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.

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

TL;DR

This work extends murmurations and Sato-Tate phenomena from rank-one to rank- zetas of elliptic curves, addressing higher-rank distributions without relying on the rank Riemann Hypothesis. It introduces a refined, RH-free framework built on sharp asymptotics for rank- invariants, notably the and associated -type observables, and demonstrates that rank Sato-Tate laws can be observed via new -distributions that align with the classical sine-squared measure. A key technical development is the recasting of rank- observations through -distributions and a direct connection to the conventional Sato-Tate law, as well as the introduction of new murmuration functionals that reveal rank stability in families of elliptic curves. The results imply that higher-rank arithmetic structures underlying the murmurations are canonically linked to the familiar abelian Sato-Tate statistics, with practical implications for understanding non-abelian zeta phenomena in arithmetic geometry.

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.
Paper Structure (8 sections, 15 theorems, 60 equations, 4 figures)

This paper contains 8 sections, 15 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rank $n$ Sato-Tate Law based on Rank $n$ Riemann Hypothesis
  3. Non-Abelian Zeta Function: Background
  4. Riemann Hypothesis in Rank $n$ for Elliptic Curves
  5. Sato-Tate Law in Rank $n$: Beyond Riemann Hypothesis
  6. Riemann Hypothesis is Too Rough
  7. Sato-Tate Law in Rank $n$: New Observing Spot
  8. Rank $n$ Murmurations for Elliptic Curves

Key Result

Theorem 1

Let ${\Bbb E}/{\Bbb Q}\space$ be a non-CM elliptic curve. Denote its $p$-reduction by $E/{\Bbb F}_p$. Then, for $\alpha, \beta\in {\Bbb R}$ satisfying $0\le \alpha <\beta \le \pi$, we have Here the big $\Delta_{E/{\Bbb F}_{p},n}$ is defined by and $\theta_{E/{\Bbb F}_{p},n}\in[0,\pi]$ is defined by

Figures (4)

  • Figure 1: Sato-Tate distribution of rank 3 zeta function $\zeta_{E/{\Bbb F}_q,3}(s)$ in terms of $\Delta'$ over elliptic curve ${\Bbb E}/{\Bbb Q}\space: y^2 +xy= x^3 + 87 x +442$ and $q \le N =10,000,000$.
  • Figure 2: Sato-Tate distribution of rank 3 zeta function $\zeta_{E/{\Bbb F}_q,3}(s)$ in terms of $\Delta"$ over elliptic curve ${\Bbb E}/{\Bbb Q}\space: y^2 +xy= x^3 + 87 x +442$ and $q \le N =10,000,000$.
  • Figure 3: Plot of $f_{r,n}^{\rm new}(i)$ where $r\in{0,1}$ and $n=5$, for elliptic curves with conductor in $[7500,10000]$. $f_{0,n}^{\rm new}(i)$ is in blue and $f_{1,n}^{\rm new}(i)$ is in red.
  • Figure 4: Plot of $f_{r,n}^{\rm new}(i)$ where $r\in{0,1}$ and $n=5$, for elliptic curves with conductor in $[7500,10000]$. $f_{0,n}^{\rm new}(i)$ is in blue and $f_{2,n}^{\rm new}(i)$ is in green.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2: Theorems 14, 15
  • Theorem 3: Theorem 17
  • Definition 4: W2
  • Theorem 5: Zeta Facts W2, see also WZ2
  • Conjecture 6: Riemann Hypothesis, W2
  • Theorem 7: Weng-Zagier WZ1
  • Theorem 8: Counting Miracle. Theorem 3 of WZ1
  • Theorem 9: Equation 6 and Theorem 3 of WZ1
  • Theorem 10: Theorem 6 of WZ1
  • ...and 11 more