The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves

TL;DR

This paper investigates the distribution of Frobenius traces in various thin parametric families of elliptic curves, extending the Sato-Tate conjecture to new, sparser parameter sets with prescribed arithmetic properties.

Contribution

It provides new results on the Sato-Tate distribution for elliptic curves with parameters in thin sets, including prime numbers and structured progressions, expanding previous work.

Findings

Distribution results for Frobenius traces in thin families

Extension of Sato-Tate conjecture to prime and structured parameter sets

Analysis of new, sparser elliptic curve families

Abstract

We obtain new results concerning the Sato-Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.