Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves II: Beyond Riemann Hypothesis (2501.10220v2)

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.

• The paper presents strong asymptotic bounds for a-invariants of non-CM elliptic curves, bypassing the traditional reliance on the Riemann Hypothesis.
• It establishes a revised rank n Sato-Tate law with novel parameters that capture higher rank zeta function distributions using refined analytical techniques.
• The findings open alternative pathways for resolving longstanding number theory conjectures and inspire innovative algorithmic approaches in computational mathematics.

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

In this paper, Shi and Weng extend their prior work on the Sato-Tate conjecture to encompass higher rank zeta functions of elliptic curves. Traditional investigations of the Sato-Tate law involve the distribution of $a$-invariants, $a_{E/\mathbb{F}_p}$, for elliptic curves without Complex Multiplication (CM) over the rationals, $\mathbb{Q}$. The authors venture beyond the classical frameworks, exploring higher rank analogues and modifying their approach to circumvent reliance on the rank $n$ Riemann Hypothesis. The investigation adopts a granular structure uncovered in earlier collaborative work involving Zagier.

Key Contributions

• Refined Bounds and Asymptotic Structures: Central to the new results are the strong asymptotic bounds for $a_{E/\mathbb{F}_p;n}$. These bounds eschew the traditional dependence on the Riemann Hypothesis for higher rank zeta functions, instead leveraging refined structures originally observed by Weng and Zagier.
• Rank $n$ Sato-Tate Law: The authors confirm that for a non-CM elliptic curve over $\mathbb{Q}$, a revised Sato-Tate law holds for higher ranks—expressly in rank $n$ distributions of $a$-invariants defined by the formula:

$$a_{E/\mathbb{F}_p;n} = (5-n) + (n-1)\cdot a_{E/\mathbb{F}_p} - (n-1)\cdot p + O\left(\frac{1}{\sqrt{p}}\right)$$

These distributions align with the classical Sato-Tate form but are founded upon stronger estimations than those traditionally leveraged.

• Novel Sato-Tate Subspaces: Utilizing the asymptotic expansions, the authors propose two new parameters, $\Theta_{E/\mathbb{F}_p;n}'$ and $\Theta_{E/\mathbb{F}_p;n}''$, which maintain the classical distribution as perceived through a fresh lens. This experimentation alludes to an intrinsic, modern alignment between higher rank structures and traditional elliptic curve behaviors.

Implications and Theoretical Developments

The paper's findings contribute profound insights into number theory and the interaction between arithmetic geometry and analytic properties of elliptic curves. The notion of a murmuration functional for elliptic curves offers elegance via functional encapsulation, capturing the intrinsic noisiness and coherence in $a$-invariant distributions across higher ranks.

The theoretical implication of processing higher rank structures absent the Riemann Hypothesis suggests alternative pathways for understanding and possibly resolving complex conjectures in number theory, without relying purely on classical conjectural frameworks.

Speculations on Future AI Developments

The methods introduced in this paper could inspire novel algorithmic techniques, especially in symbolic computation and algorithmic number theory. If effectively adapted, these insights may provide innovative means to tackle complex arithmetic problems, potentially reformulating existing approaches to longstanding conjectures.

Drawing parallels from this work, future AI systems could leverage these idiosyncratic mathematical patterns and structures to implement more advanced learning paradigms that embrace complex distributions beyond current capabilities.

In sum, Shi and Weng's research encapsulates a shift in perspective and methodology in analytic number theory, fostering nuanced understandings and breathing fresh air into the field of elliptic curve research. The paper underscores the mathematical richness inherent in high rank analyses and urges continued endeavors at the interface of arithmetic and geometry without undue reliance on conjectural assumptions.