What question did this study set out to answer?

This analysis aims to explore the Sato-Tate Conjecture concerning the distribution of Frobenius angles in elliptic curves using the Temporal Dynamics Framework.

April 15, 2026Open Access

The Sato-Tate Conjecture: Computational Analysis Through Temporal Dynamics Framework

Key Points

This analysis aims to explore the Sato-Tate Conjecture concerning the distribution of Frobenius angles in elliptic curves using the Temporal Dynamics Framework.
Applied the Temporal Dynamics Framework to analyze elliptic curves.
Compared CM and non-CM elliptic curves with focus on Frobenius angles.
Examined resonance function behavior and decoherence events.
Investigated convergence patterns and phi-to-pi transitions.
Confirmed that normalized Frobenius traces exhibit a semicircular distribution for non-CM curves.
Identified significant differences in behavior between CM and non-CM curves.
Illustrated resonance and decoherence insights through diagnostic figures.

Abstract

This paper applies the Temporal Dynamics Framework to the Sato-Tate Conjecture on the distribution of Frobenius angles of elliptic curves. The Sato-Tate Conjecture, proved for non-CM elliptic curves over Q by Taylor et al. (2011), predicts that normalised Frobenius traces follow a semicircular distribution. The Temporal Dynamics Framework analysis examines CM versus non-CM curve comparison, resonance function behaviour, decoherence events, convergence patterns, and phi-to-pi transitions. Eight diagnostic figures are included. Paper 26 in the Temporal Dynamics Framework Research Series.

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James Norman Ibbotson (Mon,) studied this question.

synapsesocial.com/papers/69df2cb9e4eeef8a2a6b1fa8 https://doi.org/https://doi.org/10.5281/zenodo.19544231

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